Effects of additional food availability and pulse control on the dynamics of a Holling-($ p $+1) type pest-natural enemy model

Xinrui Yan; Yuan Tian; Kaibiao Sun; Xinrui Yan; Yuan Tian; Kaibiao Sun

doi:10.3934/era.2023327

Electronic Research Archive

2023, Volume 31, Issue 10: 6454-6480. doi: 10.3934/era.2023327

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Research article

Effects of additional food availability and pulse control on the dynamics of a Holling-( $p$ +1) type pest-natural enemy model

1.
School of Science, Dalian Maritime University, Dalian 116026, China
2.
School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China

Received: 14 June 2023 Revised: 14 September 2023 Accepted: 20 September 2023 Published: 28 September 2023

In this paper, a novel pest-natural enemy model with additional food source and Holling-( $p$ +1) type functional response is put forward for plant pest management by considering multiple food sources for predators. The dynamical properties of the model are investigated, including existence and local asymptotic stability of equilibria, as well as the existence of limit cycles. The inhibition of natural enemy on pest dispersal and the impact of additional food sources on system dynamics are elucidated. In view of the fact that the inhibitory effect of the natural enemy on pest dispersal is slow and in general deviated from the expected target, an integrated pest management model is established by regularly releasing natural enemies and spraying insecticide to improve the control effect. The influence of the control period on the global stability and system persistence of the pest extinction periodic solution is discussed. It is shown that there exists a time threshold, and as long as the control period does not exceed that threshold, pests can be completely eliminated. When the control period exceeds that threshold, the system can bifurcate the supercritical coexistence periodic solution from the pest extinction one. To illustrate the main results and verify the effectiveness of the control method, numerical simulations are implemented in MATLAB programs. This study not only enriched the related content of population dynamics, but also provided certain reference for the management of plant pest.

Keywords:

Citation: Xinrui Yan, Yuan Tian, Kaibiao Sun. Effects of additional food availability and pulse control on the dynamics of a Holling-( $p$ +1) type pest-natural enemy model[J]. Electronic Research Archive, 2023, 31(10): 6454-6480. doi: 10.3934/era.2023327

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Abstract

1. Introduction

Einstein metrics are pivotal in numerous domains of mathematical physics and differential geometry. They are also of interest in pure mathematics, particularly in the fields of geometric analysis and algebraic geometry. An Einstein metric can be regarded as a fixed solution (up to diffeomorphism and scaling) of the Hamilton Ricci flow equation. On a Riemannian manifold $(M, g)$ , a Ricci soliton is said to exist when there is a smooth vector field $X$ and a constant $\lambda$ in the reals that satisfy the condition stated below:

$\begin{equation*} R_{ij}+\frac{1}{2}(\mathcal{L}_{X}g)_{ij} = \lambda g_{ij}, \end{equation*}$

where $R_{ij}$ is the Ricci curvature tensor, and $\mathcal{L}_{X}g$ stands for the Lie derivative of $g$ with respect to the vector field $X$ . This concept was introduced by Hamilton in ^[1], and later utilized by Perelman in his proof of the long-standing Poincare conjecture ^[2]. Lauret further generalized the notion of Einstein metrics to algebraic Ricci solitons in the Riemannian context, introducing them as a natural extension in ^[3]. Subsequently, Onda and Batat applied this framework to pseudo-Riemannian Lie groups, achieving a complete classification of algebraic Ricci solitons in three-dimensional Lorentzian Lie groups in ^[4]. Additionally, they proved that within the framework of pseudo-Riemannian manifolds, there is algebraic Ricci solitons that are not of the conventional Ricci soliton type.

In ^[5], Etayo and Santamaria explored the concept of distinguished connections on $(J^{2} = \pm1)$ -metric manifolds. This sparked interest among mathematicians in studying Ricci solitons linked to various affine connections, which can be found in ^[6,7,8]. The Bott connection was introduced in earlier works (see^[9,10,11]). In ^[12], the authors developed a theory on geodesic variations under metric changes in a geodesic foliation, with the Bott connection serving as the primary natural connection respecting the foliation's structure. In ^[13], Wu and Wang studied affine Ricci solitons and quasi-statistical structures on three-dimensional Lorentzian Lie groups associated with the Bott connection. Furthermore, in ^[14,15], the authors examined the algebraic schouten solitons and affine Ricci solitons concerning various affine connections.

Inspired by Lauret's work, Wears introduced algebraic T-solitons, linking them to T-solitons in ^[16]. Later, in ^[7], Azami introduced Schouten solitons, as a new type of generalized Ricci soliton. In this paper, I focus on algebraic Schouten solitons concerning the Bott connection with three distributions, aiming to classify and describe them on three-dimensional Lorentzian Lie groups.

In Section 2, I introduce the fundamental notions associated with three-dimensional Lie groups and algebraic Schouten soliton. In Sections 3–5, I discuss and present algebraic Schouten solitons concerning the Bott connection on three-dimensional Lorentzian Lie groups, each focusing on a different type of distribution.

2. Preliminaries

In ^[17], Milnor conducted a survey of both classical and recent findings on left-invariant Riemannian metrics on Lie groups, particularly on three-dimensional unimodular Lie groups. Furthermore, in ^[18], Rahmani classified three-dimensional unimodular Lie groups in the Lorentzian context. The non-unimodular cases were handled in ^[19,20]. Throughout this paper, I use $\{G_{i}\}_{i = 1}^{7}$ for connected, simply connected three-dimensional Lie groups equipped with a left-invariant Lorentzian metric $g$ . Their corresponding Lie algebras are denoted by $\{\mathfrak{g_{i}}\}_{i = 1}^{7}$ , and each possess a pseudo-orthonormal basis $\{e_{1}, e_{2}, e_{3}\}$ (with $e_{3}$ timelike, see ^[4]). Let $\nabla^{LC}$ and $R^{LC}$ denote the Levi-Civita connection and curvature tensor of $G_{i}$ , respectively, then

$\begin{equation*} R^{LC}(X,Y)Z = \nabla^{LC}_{X}\nabla^{LC}_{Y}Z-\nabla^{LC}_{Y}\nabla^{LC}_{X}Z-\nabla^{LC}_{[X,Y]}Z. \end{equation*}$

The Ricci tensor of $(G_{i}, g)$ is defined as follows:

$\begin{equation*} \rho^{LC}(X,Y) = -g(R^{LC}(X,e_{1})Y,e_{1})-g(R^{LC}(X,e_{2})Y,e_{2})+g(R^{LC}(X,e_{3})Y,e_{3}). \end{equation*}$

Moreover, I have the expression for the Ricci operator $Ric^{LC}$ :

$\begin{equation*} \rho^{LC}(X,Y) = g(Ric^{LC}(X),Y). \end{equation*}$

Next, recall the Bott connection, denoted by $\nabla^{B}$ . Consider a smooth manifold $(M, g)$ that is equipped with the Levi-Civita connection $\nabla$ , and let $TM$ represent its tangent bundle, spanned by $\{e_{1}, e_{2}, e_{3}\}$ . I introduce a distribution $D$ spanned by $\{e_{1}, e_{2}\}$ and its orthogonal complement $D^{\perp}$ , which is spanned by $\{e_{3}\}$ . The Bott connection $\nabla^{B}$ associated with distribution $D$ is then defined as follows:

$\begin{equation} \nabla_{X}^{B}Y = \left\{ \begin{aligned} &\pi_{D}(\nabla^{LC}_{X}Y),\quad X,Y\in\Gamma^{\infty}(D),\\ &\pi_{D}([X,Y]),\quad X\in\Gamma^{\infty}(D^{\perp}),\; Y\in\Gamma^{\infty}(D),\\ &\pi_{D^{\perp}}([X,Y]),\quad X\in\Gamma^{\infty}(D),\; Y\in\Gamma^{\infty}(D^{\perp}),\\ &\pi_{D^{\perp}}(\nabla^{LC}_{X}Y),\quad X,Y\in\Gamma^{\infty}(D^{\perp}), \end{aligned} \right. \end{equation}$

(2.1)

where $\pi_{D}$ (respectively, $\pi_{D^{\perp}}$ ) denotes the projection onto $D$ (respectively, $D^{\perp}$ ). For a more detailed discussion, refer to ^[9,10,11,21]. I denote the curvature tensor of the Bott connection $\nabla^{B}$ by $R^{B}$ , which is defined as follows:

$\begin{equation} R^{B}(X,Y)Z = \nabla^{B}_{X}\nabla^{B}_{Y}Z-\nabla^{B}_{Y}\nabla^{B}_{X}Z-\nabla^{B}_{[X,Y]}Z. \end{equation}$

(2.2)

The Ricci tensor $\rho^{B}$ associated to the connection $\nabla^{B}$ is defined as:

$\begin{equation*} \rho^{B}(X,Y) = \frac{B(X,Y)+B(Y,X)}{2}, \end{equation*}$

where

$\begin{equation*} B(X,Y) = -g(R^{B}(X,e_{1})Y,e_{1})-g(R^{B}(X,e_{2})Y,e_{2})+g(R^{B}(X,e_{3})Y,e_{3}). \end{equation*}$

Using the Ricci tensor $\rho^{B}$ , the Ricci operator $Ric^{B}$ is given by:

$\begin{equation} \rho^{B}(X,Y) = g(Ric^{B}(X),Y). \end{equation}$

(2.3)

Then, I have the definition of the Schouten tensor as follows:

$\begin{equation*} S^{B}(e_{i},e_{j}) = \rho^{B}(e_{i},e_{j})-\frac{s^{B}}{4}g(e_{i},e_{j}), \end{equation*}$

where $s^{B}$ represents the scalar curvature. Moreover, I generalized the Schouten tensor to:

$\begin{equation*} S^{B}(e_{i},e_{j}) = \rho^{B}(e_{i},e_{j})-s^{B}\lambda_{0}g(e_{i},e_{j}), \end{equation*}$

where $\lambda_{0}$ is a real number. By ^[22], I obtain the expression of $s^{B}$ as

$\begin{equation*} s^{B} = \rho^{B}(e_{1},e_{1})+\rho^{B}(e_{2},e_{2})-\rho^{B}(e_{3},e_{3}). \end{equation*}$

Definition 1. A manifold $(G_{i}, g)$ is called an algebraic Schouten soliton with associated to the connection $\nabla^{B}$ if it satisfies:

$\begin{equation*} Ric^{B} = (s^{B}\lambda_{0}+c)Id+D^{B}, \end{equation*}$

where $c$ is a constant, and $D^{B}$ is a derivation of $\mathfrak{g}_{i}$ , i.e.,

$\begin{equation} D^{B}[X_{1},X_{2}] = [D^{B}X_{1},X_{2}]+[X_{1},D^{B}X_{2}],\quad for\; X_{1},X_{2}\in \mathfrak{g}_{i}. \end{equation}$

(2.4)

3. Algebraic Schouten soliton concerning connection $\nabla^{B}$

In this section, I derive the algebraic criterion for the three-dimensional Lorentzian Lie group to exhibit an algebraic Schouten soliton related to the connection $\nabla^{B}$ . Moreover, I indicate that $G_{4}$ and $G_{7}$ do not have such solitons.

3.1. Algebraic Schouten soliton of $G_{1}$

According to ^[4], I have the expression for $\mathfrak{g}_{1}$ :

$\begin{equation*} [e_{1},e_{2}] = \alpha e_{1}-\beta e_{3},\quad [e_{1},e_{3}] = -\alpha e_{1}-\beta e_{2},\quad [e_{2},e_{3}] = \beta e_{1}+\alpha e_{2}+\alpha e_{3}, \end{equation*}$

where $\alpha\neq0$ . From this, I derive the following theorem.

Theorem 2. If $(G_{1}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B}$ ; then, it fulfills the conditions: $\beta = 0$ and $c = -\frac{1}{2}\alpha^{2}+2(\alpha^{2}+\beta^{2})\lambda_{0}$ .

Proof. From ^[23], the expression for $Ric^{B}$ is as follows:

$\begin{equation*} Ric^{B}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -(\alpha^{2}+\beta^{2}) & \alpha\beta & \frac{1}{2}\alpha\beta\\ \alpha\beta & -(\alpha^{2}+\beta^{2}) & -\frac{1}{2}\alpha^{2}\\ -\frac{1}{2}\alpha\beta & \frac{1}{2}\alpha^{2} & 0 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B} = -2(\alpha^{2}+\beta^{2})$ . Now, I can express $D^{B}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B}e_{1} = -(\alpha^{2}+\beta^{2}+s\lambda_{0}+c)e_{1}+\alpha\beta e_{2}+\frac{\alpha\beta}{2}e_{3},\\ &D^{B}e_{2} = \alpha\beta e_{1}-(\alpha^{2}+\beta^{2}+s\lambda_{0}+c)e_{2}-\frac{\alpha^{2}}{2}e_{3},\\ &D^{B}e_{3} = -\frac{\alpha\beta}{2}e_{1}+\frac{\alpha^{2}}{2}e_{2}+(2(\alpha^{2}+\beta^{2})\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B}$ on $(G_{1}, g)$ , if and only if the following condition satisfies:

$\begin{equation*} \left\{ \begin{aligned} &\frac{5}{2}\alpha^{2}\beta+2\beta^{3}-2\lambda_{0}\beta(\alpha^{2}+\beta^{2})+c\beta = 0,\\ &\frac{\alpha^{3}}{2}+2\alpha\beta^{2}-2\lambda_{0}\alpha(\alpha^{2}+\beta^{2})+c\alpha = 0,\\ &\alpha^{2}\beta = 0,\\ &2\alpha^{2}\beta+\beta^{3}-4\lambda_{0}\beta(\alpha^{2}+\beta^{2})+2c\beta = 0,\\ &\frac{\alpha^{3}}{2}+\frac{3}{2}\alpha\beta^{2}-2\lambda_{0}\alpha(\alpha^{2}+\beta^{2})+c\alpha = 0. \end{aligned} \right. \end{equation*}$

Since $\alpha\neq0$ , I have $\beta = 0$ and $c = -\frac{1}{2}\alpha^{2}+2(\alpha^{2}+\beta^{2})\lambda_{0}$ . □

3.2. Algebraic Schouten soliton of $G_{2}$

According to ^[4], I have the expression for $\mathfrak{g}_{2}$ :

$\begin{equation*} [e_{1},e_{2}] = \gamma e_{2}-\beta e_{3},\quad [e_{1},e_{3}] = -\beta e_{2}-\gamma e_{3},\quad [e_{2},e_{3}] = \alpha e_{1}, \end{equation*}$

where $\gamma\neq0$ . From this, I derive the following theorem.

Theorem 3. If $(G_{2}, g)$ constitutes an algebraic Schouten soliton concerning the connection $\nabla^{B}$ , then it fulfills the conditions: $\alpha = 0$ and $c = -\beta^{2}-\gamma^{2}+(\beta^{2}+2\gamma^{2})\lambda_{0}$ .

Proof. From ^[23], the expression for $Ric^{B}$ is as follows:

$\begin{equation*} Ric^{B}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -(\beta^{2}+\gamma^{2}) & 0 & 0\\ 0 & -(\gamma^{2}+\alpha\beta) & \frac{\alpha\gamma}{2}\\ 0 & -\frac{\alpha\gamma}{2} & 0 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B} = -(\beta^{2}+2\gamma^{2}+\alpha\beta)$ . Now, I can express $D^{B}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B}e_{1} = -(\beta^{2}+\gamma^{2}-(\beta^{2}+2\gamma^{2}+\alpha\beta)\lambda_{0}+c)e_{1},\\ &D^{B}e_{2} = -(\gamma^{2}+\alpha\beta-(\beta^{2}+2\gamma^{2}+\alpha\beta)\lambda_{0}+c)e_{2}+\frac{\alpha\gamma}{2}e_{3},\\ &D^{B}e_{3} = -\frac{\alpha\gamma}{2}e_{2}+((\beta^{2}+2\gamma^{2}+\alpha\beta)\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B}$ on $(G_{2}, g)$ , if and only if the following condition satisfies:

$\begin{equation*} \left\{ \begin{aligned} &\alpha\gamma^{2}-\beta^{3}+\alpha\beta^{2}+(\beta^{2}+2\gamma^{2}+\alpha\beta)\beta\lambda_{0}-c\beta = 0,\\ &\gamma(\beta^{2}+\gamma^{2}+\alpha\beta-(\beta^{2}+2\gamma^{2}+\alpha\beta)\lambda_{0}+c) = 0,\\ &\alpha\gamma^{2}-\beta^{3}-2\beta\gamma^{2}-\alpha\beta^{2}+(\beta^{2}+2\gamma^{2}+\alpha\beta)\beta\lambda_{0}-c\beta = 0,\\ &\alpha(-\beta^{2}+\alpha\beta-(\beta^{2}+2\gamma^{2}+\alpha\beta)\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation*}$

Suppose that $\alpha = 0$ , then $c = -\beta^{2}-\gamma^{2}+(\beta^{2}+2\gamma^{2})\lambda_{0}$ . If $\alpha\neq0$ , I have

$\begin{equation*} \left\{ \begin{aligned} &\beta(\beta^{2}+\gamma^{2}-(\beta^{2}+2\gamma^{2}+\alpha\beta)\lambda_{0}+c) = 0,\\ &\beta^{2}+\gamma^{2}+\alpha\beta-(\beta^{2}+2\gamma^{2}+\alpha\beta)\lambda_{0}+c = 0,\\ &-\beta^{2}+\alpha\beta-(\beta^{2}+2\gamma^{2}+\alpha\beta)\lambda_{0}+c = 0. \end{aligned} \right. \end{equation*}$

Since $\gamma\neq0$ , solving the equations of the above system gives $2\beta^{2}+\gamma^{2} = 0$ , which is a contradiction. □

3.3. Algebraic Schouten soliton of $G_{3}$

According to ^[4], I have the expression for $\mathfrak{g}_{3}$ :

$\begin{equation*} [e_{1},e_{2}] = -\gamma e_{3},\quad [e_{1},e_{3}] = -\beta e_{2},\quad [e_{2},e_{3}] = \alpha e_{1}. \end{equation*}$

From this, I derive the following theorem.

Theorem 4. If $(G_{3}, g)$ is an algebraic Schouten soliton concerning connection $\nabla^{B}$ ; then, one of the following cases holds:

i. $\alpha = \beta = \gamma = 0$ , for all $c$ ;

ii. $\alpha\neq0$ , $\beta = \gamma = 0$ , $c = 0$ ;

iii. $\alpha = \gamma = 0$ , $\beta\neq0$ , $c = 0$ ;

iv. $\alpha\neq0$ , $\beta\neq0$ , $\gamma = 0$ , $c = 0$ ;

v. $\alpha = \beta = 0$ , $\gamma\neq0$ , $c = 0$ ;

vi. $\alpha\neq0$ , $\beta = 0$ , $\gamma\neq0$ , $c = -\alpha\gamma+\alpha\gamma\lambda_{0}$ ;

vii. $\alpha = 0$ , $\beta\neq0$ , $\gamma\neq0$ , $c = -\beta\gamma+\beta\gamma\lambda_{0}$ .

Proof. From ^[23], the expression for $Ric^{B}$ is as follows:

$\begin{equation*} Ric^{B}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -\beta\gamma & 0 & 0\\ 0 & -\alpha\gamma & 0\\ 0 & 0 & 0 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B} = -\gamma(\alpha+\beta)$ . Now, I can express $D^{B}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B}e_{1} = -(\beta\gamma-\gamma(\alpha+\beta)\lambda_{0}+c)e_{1},\\ &D^{B}e_{2} = -(\alpha\gamma-\gamma(\alpha+\beta)\lambda_{0}+c)e_{2},\\ &D^{B}e_{3} = (\gamma(\alpha+\beta)\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B}$ on $(G_{3}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\beta\gamma^{2}+\alpha\gamma^{2}-\gamma^{2}(\alpha+\beta)\lambda_{0}+c\gamma = 0,\\ &\beta^{2}\gamma-\beta\gamma(\alpha+\beta)\lambda_{0}+c\beta = 0,\\ &\alpha^{2}\gamma-\alpha\beta\gamma-\alpha\gamma(\alpha+\beta)\lambda_{0}+c\alpha = 0. \end{aligned} \right. \end{equation}$

(3.1)

Assuming that $\gamma = 0$ . In this case, (3.1) becomes:

$\begin{equation} \left\{ \begin{aligned} &\beta c = 0,\\ &\alpha c = 0. \end{aligned} \right. \end{equation}$

(3.2)

If $\beta = 0$ , for Cases i and ii, system (3.1) holds. If $\beta\neq0$ , for Cases iii and iv, system (3.1) holds. Then, I assume that $\gamma\neq0$ . Thus, system (3.1) becomes:

$\begin{equation} \left\{ \begin{aligned} &\beta\gamma+\alpha\gamma-\gamma(\alpha+\beta)\lambda_{0}+c,\\ &\alpha \beta = 0. \end{aligned} \right. \end{equation}$

(3.3)

If $\beta = 0$ , I have Cases v and vi. If $\beta\neq0$ , for Case vii, system (3.1) holds □

3.4. Algebraic Schouten soliton of $G_{4}$

According to ^[4], $\mathfrak{g}_{4}$ takes the following form:

$\begin{equation*} [e_{1},e_{2}] = -e_{2}+(2\eta-\beta)e_{3},\quad [e_{1},e_{3}] = e_{3}-\beta e_{2},\quad [e_{2},e_{3}] = \alpha e_{1}, \end{equation*}$

where $\eta = 1\; or\; -1$ . From this, I derive the following theorem.

Theorem 5. $(G_{4}, g)$ is not an algebraic Schouten soliton concerning connection $\nabla^{B}$ .

Proof. According to ^[23], the expression for $Ric^{B}$ is derived as follows:

$\begin{equation*} Ric^{B}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -(\beta-\eta)^{2} & 0 & 0\\ 0 & 2\alpha\eta-\alpha\beta-1 & -\frac{1}{2}\alpha\\ 0 & \frac{1}{2}\alpha & 0 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B} = -((\beta-\eta)^{2}+\alpha\beta-2\alpha\eta+1)$ . Now, I can express $D^{B}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B}e_{1} = -((\beta-\eta)^{2}-((\beta-\eta)^{2}+\alpha\beta-2\alpha\eta+1)\lambda_{0}+c)e_{1},\\ &D^{B}e_{2} = (2\alpha\eta-\alpha\beta-1+((\beta-\eta)^{2}+\alpha\beta-2\alpha\eta+1)\lambda_{0}-c)e_{2}-\frac{a}{2} e_{3},\\ &D^{B}e_{3} = \frac{\alpha}{2}e_{2}+(((\beta-\eta)^{2}+\alpha\beta-2\alpha\eta+1)\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B}$ on $(G_{4}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &-(2\eta-\beta)((\beta-\eta)^{2}-2\alpha\eta+\alpha\beta+1-((\beta-\eta)^{2}+\alpha\beta-2\alpha\eta+1)\lambda_{0}+c) = \alpha,\\ &\beta((\beta-\eta)^{2}+2\alpha\eta-\alpha\beta-1-((\beta-\eta)^{2}+\alpha\beta-2\alpha\eta+1)\lambda_{0}+c) = \alpha,\\ &(\beta-\eta)^{2}-((\beta-\eta)^{2}+\alpha\beta-2\alpha\eta+1)\lambda_{0}+c = \alpha(\eta-\beta),\\ &\alpha((\beta-\eta)^{2}+2\alpha\eta-\alpha\beta-1+((\beta-\eta)^{2}+\alpha\beta-2\alpha\eta+1)\lambda_{0}-c) = 0. \end{aligned} \right. \end{equation}$

(3.4)

I now analyze the system under different assumptions.

Assuming that $\alpha = 0$ . Then, system (3.4) becomes:

$\begin{equation*} \left\{ \begin{aligned} &(2\eta-\beta)((\beta-\eta)^{2}+1-((\beta-\eta)^{2}+1)\lambda_{0}+c) = 0,\\ &\beta((\beta-\eta)^{2}-1-((\beta-\eta)^{2}+1)\lambda_{0}+c) = 0,\\ &(\beta-\eta)^{2}-((\beta-\eta)^{2}+1)\lambda_{0}+c = 0. \end{aligned} \right. \end{equation*}$

Upon direct calculation, it is evident that $(2\eta-\beta) = \beta = 0$ , which leads to a contradiction.

If $\alpha\neq0$ , we have

$\begin{equation} \left\{ \begin{aligned} &(2\eta-\beta)(\alpha\eta-1) = \alpha,\\ &\beta(3\alpha\eta-2\alpha\beta-1) = \alpha,\\ &(\beta-\eta)^{2}-\alpha(\eta-\beta)-((\beta-\eta)^{2}-2\alpha\eta+\alpha\beta+1)\lambda_{0}+c = 0,\\ &\alpha((\beta-\eta)^{2}+2\alpha\eta-\alpha\beta-1+((\beta-\eta)^{2}-2\alpha\eta+\alpha\beta+1)\lambda_{0}-c) = 0. \end{aligned} \right. \end{equation}$

(3.5)

From the last two equations in (3.5), we have $(\beta-\eta)^{2} = (1-\alpha\eta)$ . Substituting into the first two equations in (3.5) yields $\alpha = \eta$ , which is a contradiction. Therefore, system (3.4) has no solutions. Then, the theorem is true. □

3.5. Algebraic Schouten soliton of $G_{5}$

According to ^[4], we have the expression for $\mathfrak{g}_{5}$ :

$\begin{equation*} [e_{1},e_{2}] = 0,\quad [e_{1},e_{3}] = \alpha e_{1}+\beta e_{2},\quad [e_{2},e_{3}] = \gamma e_{1}+\delta e_{2}, \end{equation*}$

where $\alpha+\delta\neq0$ and $\alpha\gamma-\beta\delta = 0$ . From this, we derive the following theorem.

Theorem 6. If $(G_{5}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B}$ , then $c = 0$ .

Proof. According to ^[23], the expression for $Ric^{B}$ is derived as follows:

$\begin{equation*} Ric^{B}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B} = 0$ . Now, I can express $D^{B}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B}e_{1} = -ce_{1},\\ &D^{B}e_{2} = -ce_{2},\\ &D^{B}e_{3} = -ce_{3}. \end{aligned} \right. \end{equation*}$

Hence, by (2.4), I conclude that there is an algebraic Schouten soliton associated with $\nabla^{B}$ on $(G_{5}, g)$ . Furthermore, for this algebraic Schouten soliton, I have $c = 0$ . □

3.6. Algebraic Schouten soliton of $G_{6}$

According to ^[4], I have the expression for $\mathfrak{g}_{6}$ :

$\begin{equation*} \begin{aligned} &[e_{1},e_{2}] = \alpha e_{2}+\beta e_{3},\\ &[e_{1},e_{3}] = \gamma e_{2}+\delta e_{3},\\ &[e_{2},e_{3}] = 0, \end{aligned} \end{equation*}$

where $\alpha+\delta\neq0$ and $\alpha\gamma-\beta\delta = 0$ . From this, I derive the following theorem.

Theorem 7. If $(G_{6}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B}$ , then one of the following cases holds:

i. $\alpha = \beta = \gamma = 0$ , $\delta\neq0$ , $c = 0$ ;

ii. $\alpha = \beta = 0$ , $\gamma\neq0$ , $\delta\neq0$ , $c = 0$ ;

iii. $\alpha\neq0$ , $\beta = \gamma = \delta = 0$ , $c = -\alpha^{2}+2\alpha^{2}\lambda_{0}$ ;

iv. $\alpha\neq0$ , $\beta = \gamma = 0$ , $\delta\neq0$ , $c = -\alpha^{2}+2\alpha^{2}\lambda_{0}$ .

Proof. From ^[23], I have the expression for $Ric^{B}$ as follows:

$\begin{equation*} Ric^{B}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -(\alpha^{2}+\beta\gamma)^{2} & 0 & 0\\ 0 & -\alpha^{2} & 0\\ 0 & 0 & 0 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B} = -(2\alpha^{2}+\beta\gamma)$ . Now, I can express $D^{B}$ as follows:

$\begin{equation} \left\{ \begin{aligned} &D^{B}e_{1} = -(\alpha^{2}+\beta\gamma-(2\alpha^{2}+\beta\gamma)\lambda_{0}+c)e_{1},\\ &D^{B}e_{2} = -(\alpha^{2}-(2\alpha^{2}+\beta\gamma)\lambda_{0}+c)e_{2},\\ &D^{B}e_{3} = ((2\alpha^{2}+\beta\gamma)\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation}$

(3.6)

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B}$ on $(G_{6}, g)$ , if and only if the following condition satisfies:

$\begin{equation*} \left\{ \begin{aligned} &\beta(2\alpha^{2}+\beta\gamma-(2\alpha^{2}+\beta\gamma)\lambda_{0}+c) = 0,\\ &\alpha(\alpha^{2}+\beta\gamma-(2\alpha^{2}+\beta\gamma)\lambda_{0}+c) = 0,\\ &\gamma(\beta\gamma-(2\alpha^{2}+\beta\gamma)\lambda_{0}+c) = 0,\\ &\delta(\alpha^{2}+\beta\gamma-(2\alpha^{2}+\beta\gamma)\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation*}$

From the first equation above, we have either $\beta = 0$ or $\beta\neq0$ . I now analyze the system under different assumptions.

Assuming that $\beta = 0$ , I have:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\alpha^{2}-2\alpha^{2}\lambda_{0}+c) = 0,\\ &\gamma(-2\alpha^{2}\lambda_{0}+c) = 0,\\ &\delta(\alpha^{2}-2\alpha^{2}\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(3.7)

Given $\alpha\gamma-\beta\delta = 0$ and $\alpha+\delta\neq0$ , assume first that $\alpha = 0$ . In this case, system (3.7) can be simplified to:

$\begin{equation*} s^{B}\lambda_{0}+c = 0. \end{equation*}$

Then, I have Cases i and ii. If $\beta\neq0$ , system (3.7) becomes:

$\begin{equation} \alpha^{2}+s^{B}\lambda_{0}+c = 0. \end{equation}$

(3.8)

Then, I have Cases iii and iv.

If $\beta\neq0$ , system (3.7) becomes:

$\begin{equation} \left\{ \begin{aligned} &\beta(2\alpha^{2}+\beta\gamma-(2\alpha^{2}+\beta\gamma)\lambda_{0}+c) = 0,\\ &\alpha(\alpha^{2}+\beta\gamma-(2\alpha^{2}+\beta\gamma)\lambda_{0}+c) = 0, \end{aligned} \right. \end{equation}$

(3.9)

which is a contradiction. □

3.7. Algebraic Schouten soliton of $G_{7}$

According to ^[4], I have the expression for $\mathfrak{g}_{7}$ :

$\begin{equation*} \begin{aligned} &[e_{1},e_{2}] = -\alpha e_{1}-\beta e_{2}-\beta e_{3},\\ &[e_{1},e_{3}] = \alpha e_{1}+\beta e_{2}+\beta e_{3},\\ &[e_{2},e_{3}] = \gamma e_{1}+\delta e_{2}+\delta e_{3}, \end{aligned} \end{equation*}$

where $\alpha+\delta\neq0$ and $\alpha\delta = 0$ . From this, I derive the following theorem.

Theorem 8. $(G_{7}, g)$ is not an algebraic Schouten soliton concerning connection $\nabla^{B}$ .

Proof. From ^[23], I have the expression for $Ric^{B}$ as follows:

$\begin{equation*} Ric^{B}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -\alpha^{2} & \frac{1}{2}\beta(\delta-\alpha) & -\delta(\alpha+\delta)\\ \frac{1}{2}\beta(\delta-\alpha) & -(\alpha^{2}+\beta^{2}+\beta\delta) & -\delta^{2}-\frac{1}{2}(\beta\gamma+\alpha\delta)\\ \delta(\alpha+\delta) & \delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta) & 0 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B} = -(2\alpha^{2}+\beta^{2}+\beta\delta)$ . Now, I can express $D^{B}$ as follows:

$\begin{equation*} \left\{ \begin{array}{lll} D^{B}e_{1} = -(\alpha^{2}-(2\alpha^{2}+\beta^{2}+\beta\delta)\lambda_{0}+c)e_{1}+\frac{1}{2}\beta(\delta-\alpha)e_{2}-\delta(\alpha+\delta)e_{3},\\ D^{B}e_{2} = \frac{1}{2}\beta(\delta-\alpha)e_{1}-(\alpha^{2}+\beta^{2}+\beta\delta-(2\alpha^{2}+\beta^{2}+\beta\delta)\lambda_{0}+c)e_{2}-(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta))e_{3},\\ D^{B}e_{3} = \delta(\alpha+\delta)e_{1}+(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta))e_{2}+((2\alpha^{2}+\beta^{2}+\beta\delta)\lambda_{0}-c)e_{3}.\ \end{array} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B}$ on $(G_{7}, g)$ , if and only if the following condition satisfies:

$\begin{equation*} \left\{ \begin{array}{ll} \alpha(\alpha^{2}+\beta^{2}+\beta\alpha+s^{B}\lambda_{0}+c)+(\gamma+\beta)\delta(\alpha+\delta)+\frac{1}{2}\beta^{2}(\delta-\alpha) = \alpha(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)),\\ \beta(\alpha+s^{B}\lambda_{0}+c)+\delta^{2}(\alpha+\delta)+\frac{1}{2}\alpha\beta(\delta-\alpha) = 0,\\ \beta(2\alpha^{2}+\beta^{2}+\beta\alpha+s^{B}\lambda_{0}+c)+\delta(\delta-\alpha)(\alpha+\delta) = 2\beta(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)),\\ \alpha(s^{B}\lambda_{0}+c)+\alpha(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta))+\frac{1}{2}\beta(\beta-\gamma)(\delta-\alpha)+\beta\delta(\alpha+\delta) = 0,\\ \beta(-\beta^{2}-\beta\alpha+s^{B}\lambda_{0}+c)-\frac{1}{2}\beta(\delta-\alpha)^{2}+2\beta(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)) = 0,\\ \beta(\alpha^{2}+s^{B}\lambda_{0}+c) = \alpha\delta(\alpha+\delta)+\frac{1}{2}\beta\delta(\delta-\alpha),\\ \gamma(\beta^{2}+\beta\alpha+s^{B}\lambda_{0}+c) = \delta(\alpha-\delta)(\alpha+\delta)-\frac{1}{2}\beta(\delta-\alpha)^{2},\\ \delta(s^{B}\lambda_{0}+c)+\delta(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)) = \beta\delta(\alpha+\delta)+\frac{1}{2}\beta(\delta-\alpha)(\beta-\gamma),\\ \delta(\alpha^{2}+\beta^{2}+\beta\alpha+s^{B}\lambda_{0}+c)-\delta(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)) = (\beta+\gamma)\delta(\alpha+\delta)+\frac{1}{2}\beta^{2}(\delta-\alpha). \end{array} \right. \end{equation*}$

Recall that $\alpha+\delta\neq0$ and $\alpha\gamma = 0$ . I now analyze the system under different assumptions:

Assume first that $\alpha\neq0$ , $\gamma = 0$ . Then, the above system becomes:

$\begin{equation} \left\{ \begin{array}{ll} \alpha(\alpha^{2}+\beta^{2}+\beta\alpha+s\lambda_{0}+c)+\beta\delta(\alpha+\delta)+\frac{1}{2}\beta^{2}(\delta-\alpha) = \alpha(\delta^{2}+\frac{1}{2}\alpha\delta),\\ \beta(\alpha+s\lambda_{0}+c)+\delta^{2}(\alpha+\delta)+\frac{1}{2}\alpha\beta(\delta-\alpha) = 0,\\ \beta(2\alpha^{2}+\beta^{2}+\beta\alpha+s\lambda_{0}+c)+\delta(\delta-\alpha)(\alpha+\delta) = 2\beta(\delta^{2}+\frac{1}{2}\alpha\delta),\\ \alpha(s\lambda_{0}+c)+\alpha(\delta^{2}+\frac{1}{2}\alpha\delta)+\frac{1}{2}\beta^{2}(\delta-\alpha)+\beta\delta(\alpha+\delta) = 0,\\ \beta(-\beta^{2}-\beta\alpha+s\lambda_{0}+c)-\frac{1}{2}\beta(\delta-\alpha)^{2}+2\beta(\delta^{2}+\frac{1}{2}\alpha\delta) = 0,\\ \beta(\alpha^{2}+s\lambda_{0}+c) = \alpha\delta(\alpha+\delta)+\frac{1}{2}\beta\delta(\delta-\alpha),\\ \delta(\alpha-\delta)(\alpha+\delta)-\frac{1}{2}\beta(\alpha-\delta)^{2} = 0,\\ \beta\delta(\alpha+\delta)+\frac{1}{2}\beta^{2}(\delta-\alpha) = \delta(s\lambda_{0}+c)+\delta(\delta^{2}+\frac{1}{2}\alpha\delta),\\ \beta\delta(\alpha+\delta)+\frac{1}{2}\beta^{2}(\delta-\alpha) = \delta(\alpha^{2}+\beta^{2}+\beta\alpha+s\lambda_{0}+c)-\delta(\delta^{2}+\frac{1}{2}\alpha\delta). \end{array} \right. \end{equation}$

(3.10)

Next, suppose that $\beta = 0$ , I have:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\alpha^{2}+s\lambda_{0}+c)-\alpha(\delta^{2}+\frac{1}{2}\alpha\delta) = 0,\\ &\delta^{2}(\alpha+\delta) = 0. \end{aligned} \right. \end{equation}$

(3.11)

Which is a contradiction.

If $\beta\neq0$ , we further assume that $\delta = 0$ . Under this assumption, the last equation in (3.10) yields $\alpha^{2}\beta = 0$ , which leads to a contradiction. If we presume $\alpha = \delta$ , then the equations in (3.10) imply that $\alpha\delta(\alpha+\delta) = -\delta^{2}(\alpha+\delta)$ , which is a contradiction. Additionally, if I assume that $\delta\neq0$ and $\delta\neq-\alpha$ , then from equation system (3.10), I have the following equation:

$\begin{equation} \left\{ \begin{aligned} &\alpha^{2}+s\lambda_{0}+c = \frac{(\delta-\alpha)^{2}}{2},\\ &-\beta^{2}-\alpha\beta = \frac{(\delta-\alpha)^{2}}{2}-(\delta^{2}+\frac{\alpha\delta}{2}). \end{aligned} \right. \end{equation}$

(3.12)

Substituting (3.12) into the third equation in system (3.10) yields $\alpha^{2} = -\frac{1}{2}(\delta-\alpha)^{2}$ , which is a contradiction.

Second, let $\alpha = 0$ , $\gamma\neq0$ . Then, if $\beta = 0$ , the second equation in (3.10) reduces to $\delta^{3} = 0$ , which contradicts with $\alpha+\delta\neq0$ . On the other hand, if $\beta\neq0$ , I can derive from the second and sixth equations in system (3.10) that $\delta+\frac{1}{2}\beta = 0$ . Substituting into the fourth equation in (3.10) yields $\gamma = 0$ , which is a contradiction. □

4. Algebraic Schouten solitons concerning connection $\nabla^{B_{1}}$

In this section, I formulate the algebraic criterion necessary for a three-dimensional Lorentzian Lie group to have an algebraic Schouten soliton related to the Bott connection $\nabla^{B_{1}}$ . Recall the Bott connection, denoted by $\nabla^{B_{1}}$ , with the second distribution. Consider a smooth manifold $(M, g)$ that is equipped with the Levi-Civita connection $\nabla$ , and let $TM$ represent its tangent bundle, spanned by $\{e_{1}, e_{2}, e_{3}\}$ . I introduce a distribution $D_{1}$ spanned by $\{e_{1}, e_{3}\}$ and its orthogonal complement $D^{\perp}_{1}$ , which is spanned by $\{e_{2}\}$ . The Bott connection $\nabla^{B}_{1}$ associated with the distribution $D_{1}$ is then defined as follows:

$\begin{equation*} \nabla_{X}^{B_{1}}Y = \left\{ \begin{aligned} &\pi_{D_{1}}(\nabla^{LC}_{X}Y),\quad X,Y\in\Gamma^{\infty}(D_{1}),\\ &\pi_{D_{1}}([X,Y]),\quad X\in\Gamma^{\infty}(D_{1}^{\perp}),\; Y\in\Gamma^{\infty}(D_{1}),\\ &\pi_{D_{1}^{\perp}}([X,Y]),\quad X\in\Gamma^{\infty}(D_{1}),\; Y\in\Gamma^{\infty}(D_{1}^{\perp}),\\ &\pi_{D_{1}^{\perp}}(\nabla^{LC}_{X}Y),\quad X,Y\in\Gamma^{\infty}(D_{1}^{\perp}), \end{aligned} \right. \end{equation*}$

where $\pi_{D_{1}}$ (respectively, $\pi_{D_{1}^{\perp}}$ ) denotes the projection onto $D_{1}$ (respectively, $D_{1}^{\perp}$ ).

4.1. Algebraic Schouten soliton of $G_{1}$

Lemma 9. ^[13] The Ricci tensor $\rho^{B_{1}}$ concerning connection $\nabla^{B_{1}}$ of $(G_{1}, g)$ is given by:

$\begin{equation} \rho^{B_{1}}(e_{i},e_{j}) = \left( \begin{matrix} \alpha^{2}-\beta^{2} & \frac{1}{2}\alpha\beta & -\alpha\beta\\ \frac{1}{2}\alpha\beta & 0 & \frac{1}{2}\alpha^{2}\\ -\alpha\beta & \frac{1}{2}\alpha^{2} & \beta^{2}-\alpha^{2} \end{matrix} \right). \end{equation}$

(4.1)

From this, I derive the following theorem.

Theorem 10. If $(G_{1}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{1}}$ , then it fulfills the conditions: $\beta = 0$ and $c = \frac{1}{2}\alpha^{2}-2\alpha^{2}\lambda_{0}$ .

Proof. According to (4.1), the expression for $Ric^{B_{1}}$ is derived as follows:

$\begin{equation*} Ric^{B_{1}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} \alpha^{2}-\beta^{2} & \frac{1}{2}\alpha\beta & \alpha\beta\\ \frac{1}{2}\alpha\beta & 0 & -\frac{1}{2}\alpha^{2}\\ -\alpha\beta & \frac{1}{2}\alpha^{2} & \alpha^{2}-\beta^{2} \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{1}} = 2(\alpha^{2}-\beta^{2})$ . Now, I can express $D^{B_{1}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{1}}e_{1} = (\alpha^{2}-\beta^{2}-2(\alpha^{2}-\beta^{2})\lambda_{0}-c)e_{1}+\frac{1}{2}\alpha\beta e_{2}+\alpha\beta e_{3},\\ &D^{B_{1}}e_{2} = \frac{1}{2}\alpha\beta e_{1}-(2(\alpha^{2}-\beta^{2})\lambda_{0}+c)e_{2}-\frac{\alpha^{2}}{2}e_{3},\\ &D^{B_{1}}e_{3} = -\alpha\beta e_{1}+\frac{1}{2}\alpha^{2} e_{2}+(\alpha^{2}-\beta^{2}-2(\alpha^{2}-\beta^{2})\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{1}}$ on $(G_{1}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\alpha(2(\alpha^{2}-\beta^{2})\lambda_{0}+c)+2\alpha\beta^{2}-\frac{1}{2}\alpha^{3} = 0,\\ &\alpha^{2}\beta = 0,\\ &\beta(2(\alpha^{2}-\beta^{2})\lambda_{0}+c)-2\alpha^{2}\beta = 0,\\ &\alpha(\alpha^{2}-\beta^{2}-2(\alpha^{2}-\beta^{2})\lambda_{0}-c)-\alpha\beta^{2}-\frac{1}{2}\alpha^{3} = 0,\\ &\beta(\alpha^{2}-2\beta^{2}-2(\alpha^{2}-\beta^{2})\lambda_{0}-c) = 0,\\ &\alpha(2(\alpha^{2}-\beta^{2})\lambda_{0}+c)-\frac{1}{2}\alpha^{3}+2\alpha\beta^{2} = 0. \end{aligned} \right. \end{equation}$

(4.2)

Since $\alpha\neq0$ , the second equation in (4.2) yields $\beta = 0$ . Then, I have $c = \frac{1}{2}\alpha^{2}-2\alpha^{2}\lambda_{0}$ . □

4.2. Algebraic Schouten soliton of $G_{2}$

Lemma 11. ^[13] The Ricci tensor $\rho^{B_{1}}$ concerning the connection $\nabla^{B_{1}}$ of $(G_{2}, g)$ is given by:

$\begin{equation} \rho^{B_{1}}(e_{i},e_{j}) = \left( \begin{matrix} -(\beta^{2}+\gamma^{2}) & 0 & 0\\ 0 & 0 & -\frac{1}{2}\alpha\gamma\\ 0 & -\frac{1}{2}\alpha\gamma & \alpha\beta+\gamma^{2} \end{matrix} \right). \end{equation}$

(4.3)

From this, I derive the following theorem.

Theorem 12. If $(G_{2}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{1}}$ ; then, it fulfills the conditions: $\alpha = \beta = 0$ and $c = -\gamma^{2}+\gamma^{2}\lambda_{0}$ .

Proof. According to (4.3), the expression for $Ric^{B_{1}}$ is derived as follows:

$\begin{equation*} Ric^{B_{1}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -(\beta^{2}+\gamma^{2}) & 0 & 0\\ 0 & 0 & \frac{1}{2}\alpha\gamma\\ 0 & -\frac{1}{2}\alpha\gamma & -\alpha\beta-\gamma^{2} \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{1}} = -(\beta^{2}+\gamma^{2}+\alpha\beta)$ . Now, I can express $D^{B_{1}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{1}}e_{1} = -(\beta^{2}+\gamma^{2}-(\beta^{2}+\gamma^{2}+\alpha\beta)\lambda_{0}+c)e_{1},\\ &D^{B_{1}}e_{2} = ((\beta^{2}+\gamma^{2}+\alpha\beta)\lambda_{0}-c)e_{2}+\frac{1}{2}\alpha\gamma e_{3},\\ &D^{B_{1}}e_{3} = -\frac{1}{2}\alpha\gamma e_{2}-(\alpha\beta+\gamma^{2}-(\beta^{2}+\gamma^{2}+\alpha\beta)\lambda_{0}+c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{1}}$ on $(G_{2}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\beta^{2}+\gamma^{2}-(\beta^{2}+\gamma^{2}+\alpha\beta)\lambda_{0}+c)+\frac{1}{2}\alpha\beta\gamma = 0,\\ &\beta((\beta^{2}+\gamma^{2}+\alpha\beta)\lambda_{0}-c)+\alpha\gamma^{2} = 0,\\ &\beta(\beta^{2}+2\gamma^{2}+\alpha\beta-(\beta^{2}+\gamma^{2}+\alpha\beta)\lambda_{0}+c)-\alpha\gamma^{2} = 0,\\ &\gamma(\beta^{2}+\gamma^{2}-(\beta^{2}+\gamma^{2}+\alpha\beta)\lambda_{0}+c)+\frac{1}{2}\alpha\beta\gamma = 0,\\ &\alpha(\alpha\beta-\beta^{2}-(\beta^{2}+\gamma^{2}+\alpha\beta)\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(4.4)

By solving (4.4), I get $\alpha = \beta = 0$ , $c = -\gamma^{2}+\gamma^{2}\lambda_{0}$ . □

4.3. Algebraic Schouten soliton of $G_{3}$

Lemma 13. ^[13] The Ricci tensor $\rho^{B_{1}}$ concerning connection $\nabla^{B_{1}}$ of $(G_{3}, g)$ is given by:

$\begin{equation} \rho^{B_{1}}(e_{i},e_{j}) = \left( \begin{matrix} -\beta\gamma & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & \alpha\beta \end{matrix} \right). \end{equation}$

(4.5)

From this, I derive the following theorem.

Theorem 14. If $(G_{3}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{1}}$ ; then, one of the following cases holds:

i. $\alpha = \beta = \gamma = 0$ , for all $c$ ;

ii. $\alpha\neq0$ , $\beta = \gamma = 0$ , $c = 0$ ;

iii. $\alpha = 0$ , $\beta\neq0$ , $\gamma = 0$ , $c = 0$ ;

iv. $\alpha\neq0$ , $\beta\neq0$ , $\gamma = 0$ , $c = \alpha\beta\lambda_{0}$ ;

v. $\alpha = \beta = 0$ , $\gamma\neq0$ , $c = 0$ , ;

vi. $\alpha\neq0$ , $\beta = 0$ , $\gamma\neq0$ , $c = 0$ ;

vii. $\alpha = 0$ , $\beta\neq0$ , $\gamma\neq0$ , $c = -\beta\gamma+\beta\gamma\lambda_{0}$ .

Proof. According to (4.5), the expression for $Ric^{B_{1}}$ is derived as follows:

$\begin{equation*} Ric^{B_{1}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -\beta\gamma & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -\alpha\beta \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{1}} = -(\beta\gamma+\alpha\beta)$ . Now, I can express $D^{B_{1}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{1}}e_{1} = -(\beta\gamma-(\beta\gamma+\alpha\beta)\lambda_{0}+c)e_{1},\\ &D^{B_{1}}e_{2} = ((\beta\gamma+\alpha\beta)\lambda_{0}-c)e_{2},\\ &D^{B_{1}}e_{3} = -(\alpha\beta-(\beta\gamma+\alpha\beta)\lambda_{0}+c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{1}}$ on $(G_{3}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\gamma(\beta\gamma-\alpha\beta-(\beta\gamma+\alpha\beta)\lambda_{0}+c) = 0,\\ &\beta(\alpha\beta+\beta\gamma-(\beta\gamma+\alpha\beta)\lambda_{0}+c) = 0,\\ &\alpha(\alpha\beta-\beta\gamma-(\beta\gamma+\alpha\beta)\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(4.6)

Assume first that $\gamma = 0$ ; then, (4.6) reduces to:

$\begin{equation*} \left\{ \begin{aligned} &\beta(\alpha\beta-\alpha\beta\lambda_{0}+c) = 0,\\ &\alpha(\alpha\beta-\alpha\beta\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation*}$

Then, I have Cases i–iv.

Now, let $\gamma\neq0$ . From the last two equations in (4.6), I obtain $\alpha\beta\gamma = 0$ . If $\beta = 0$ , then it follows that $c = 0$ . Therefore, Cases v and vi are valid. If $\beta\neq0$ , we deduce $c = -\beta\gamma+\beta\gamma\lambda_{0}$ ; then, for Case vii, system (4.6) holds. □

4.4. Algebraic Schouten soliton of $G_{4}$

Lemma 15. ^[13] The Ricci tensor $\rho^{B_{1}}$ concerning connection $\nabla^{B_{1}}$ of $(G_{4}, g)$ is given by:

$\begin{equation} \rho^{B_{1}}(e_{i},e_{j}) = \left( \begin{matrix} -(\beta-\eta)^{2} & 0 & 0\\ 0 & 0 & \frac{1}{2}\alpha\\ 0 & \frac{1}{2}\alpha & \alpha\beta+1 \end{matrix} \right). \end{equation}$

(4.7)

From this, I derive the following theorem.

Theorem 16. If $(G_{4}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{1}}$ ; then, it fulfills the conditions: $\beta = \eta$ , $\alpha = -\beta$ and $c = 0$ .

Proof. According to (4.7), the expression for $Ric^{B_{1}}$ is derived as follows:

$\begin{equation*} Ric^{B_{1}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -(\beta-\eta)^{2} & 0 & 0\\ 0 & 0 & -\frac{1}{2}\alpha\\ 0 & \frac{1}{2}\alpha & -\alpha\beta-1 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{1}} = -((\beta-\eta)^{2}+\alpha\beta+1)$ . Now, I can express $D^{B_{1}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{1}}e_{1} = -((\beta-\eta)^{2}-((\beta-\eta)^{2}+\alpha\beta+1)\lambda_{0}+c)e_{1},\\ &D^{B_{1}}e_{2} = (((\beta-\eta)^{2}+\alpha\beta+1)\lambda_{0}-c)e_{2}-\frac{1}{2}\alpha e_{3},\\ &D^{B_{1}}e_{3} = \frac{1}{2}\alpha-(\alpha\beta+1-((\beta-\eta)^{2}+\alpha\beta+1)\lambda_{0}+c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{1}}$ on $(G_{4}, g)$ , if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &(\beta-\eta)^{2}-((\beta-\eta)^{2}+\alpha\beta+1)\lambda_{0}+c-\alpha(\eta-\beta) = 0,\\ &(2\eta-\beta)((\beta-\eta)^{2}-\alpha\beta-1-((\beta-\eta)^{2}+\alpha\beta+1)\lambda_{0}+c)+\alpha = 0,\\ &\beta((\beta-\eta)^{2}+\alpha\beta+1-((\beta-\eta)^{2}+\alpha\beta+1)\lambda_{0}+c)-\alpha = 0,\\ &\alpha(\alpha\beta+1-(\beta-\eta)^{2}-((\beta-\eta)^{2}+\alpha\beta+1)\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(4.8)

By solving the above system, I obtain the solutions $\beta = \eta$ , $\alpha = -\beta$ and $c = 0$ . In this case, the theorem is true. □

4.5. Algebraic Schouten soliton of $G_{5}$

Lemma 17. ^[13] The Ricci tensor $\rho^{B_{1}}$ concerning connection $\nabla^{B_{1}}$ of $(G_{5}, g)$ is given by:

$\begin{equation} \rho^{B_{1}}(e_{i},e_{j}) = \left( \begin{matrix} \alpha^{2} & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -(\beta\gamma+\alpha^{2}) \end{matrix} \right). \end{equation}$

(4.9)

From this, I derive the following result.

Theorem 18. If $(G_{5}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{1}}$ ; then, one of the following cases holds:

i. $\alpha = \beta = \gamma = 0$ , $c = 0$ ;

ii. $\alpha = \beta = 0$ , $\gamma\neq0$ , $c = 0$ ;

iii. $\alpha = 0$ , $\beta\neq0$ , $\gamma\neq0$ , $c = -\beta\gamma+\beta\gamma\lambda_{0}$ ;

iv. $\alpha\neq0$ , $\beta = \delta = \gamma = 0$ , $c = -\alpha^{2}-2\alpha^{2}\lambda_{0}$ ;

v. $\alpha\neq0$ , $\beta = \gamma = 0$ , $\delta\neq0$ , $c = \alpha^{2}-2\alpha^{2}\lambda_{0}$ .

Proof. From (4.9), I have the expression for $Ric^{B_{1}}$ as follows:

$\begin{equation*} Ric^{B_{1}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} \alpha^{2} & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & (\beta\gamma+\alpha^{2}) \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{1}} = (2\alpha^{2}+\beta\gamma)$ . Now, I can express $D^{B_{1}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{1}}e_{1} = (\alpha^{2}-(2\alpha^{2}+\beta\gamma)\lambda_{0}-c),\\ &D^{B_{1}}e_{2} = -((2\alpha^{2}+\beta\gamma)\lambda_{0}+c)e_{2}\\ &D^{B_{1}}e_{3} = (\beta\gamma+\alpha^{2}-(2\alpha^{2}+\beta\gamma)\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there exists an algebraic Schouten soliton associated to $\nabla^{B_{1}}$ on $(G_{5}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\beta\gamma+\alpha^{2}-(\beta\gamma+2\alpha^{2})\lambda_{0}-c) = 0,\\ &\beta(\beta\gamma+2\alpha^{2}-(\beta\gamma+2\alpha^{2})\lambda_{0}-c) = 0,\\ &\gamma(\beta\gamma-(\beta\gamma+2\alpha^{2})\lambda_{0}-c) = 0,\\ &\delta(\beta\gamma+\alpha^{2}-(\beta\gamma+2\alpha^{2})\lambda_{0}-c) = 0. \end{aligned} \right. \end{equation}$

(4.10)

Assume first that $\alpha = 0$ , so I have:

$\begin{equation} \left\{ \begin{aligned} &\beta(\beta\gamma-\beta\gamma\lambda_{0}-c) = 0,\\ &\gamma(\beta\gamma-\beta\gamma\lambda_{0}-c) = 0,\\ &\delta(\beta\gamma-\beta\gamma\lambda_{0}-c) = 0. \end{aligned} \right. \end{equation}$

(4.11)

Then, for Cases i–iii, system (4.10) holds.

Now, I let $\alpha\neq0$ . The second equation in (4.10) leads to $\beta = 0$ , then system (4.10) reduces to:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\alpha^{2}-2\alpha^{2}\lambda_{0}-c) = 0,\\ &\gamma(2\alpha^{2}\lambda_{0}+c) = 0,\\ &\delta(\alpha^{2}-2\alpha^{2}\lambda_{0}-c) = 0. \end{aligned} \right. \end{equation}$

(4.12)

This proves that Cases iv and v hold. □

4.6. Algebraic Schouten soliton of $G_{6}$

Lemma 19. ^[13] The Ricci tensor $\rho^{B_{1}}$ concerning connection $\nabla^{B_{1}}$ of $(G_{6}, g)$ is given by:

$\begin{equation} \rho^{B_{1}}(e_{i},e_{j}) = \left( \begin{matrix} -(\delta^{2}+\beta\gamma) & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & \delta^{2} \end{matrix} \right). \end{equation}$

(4.13)

From this, I derive the following result.

Theorem 20. If $(G_{6}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{1}}$ ; then, one of the following cases holds:

1) $\alpha = \beta = \gamma = 0$ , $\delta\neq0$ , $c = -\delta^{2}+2\delta^{2}\lambda_{0}$ ;

2) $\alpha\neq0$ , $\beta = \delta = \gamma = 0$ , $c = 0$ ;

3) $\alpha\neq0$ , $\beta\neq0$ , $\delta = \gamma = 0$ , $c = 0$ ;

4) $\alpha\neq0$ , $\beta = \gamma = 0$ , $\delta\neq0$ , $c = -\delta^{2}+2\delta^{2}\lambda_{0}$ .

Proof. From (4.13), I have the expression for $Ric^{B_{1}}$ as follows:

$\begin{equation*} Ric^{B_{1}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} -(\delta^{2}+\beta\gamma) & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -\delta^{2} \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{1}} = -(2\delta^{2}+\beta\gamma)$ . Now, I can express $D^{B_{1}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{1}}e_{1} = -(\delta^{2}+\beta\gamma-(2\delta^{2}+\beta\gamma)\lambda_{0}+c)e_{1},\\ &D^{B_{1}}e_{2} = ((2\delta^{2}+\beta\gamma)\lambda_{0}-c)e_{2},\\ &D^{B_{1}}e_{3} = -(\delta^{2}-(2\delta^{2}+\beta\gamma)\lambda_{0}+c). \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{1}}$ on $(G_{6}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\delta^{2}+\beta\gamma-(2\delta^{2}+\beta\gamma)\lambda_{0}+c) = 0,\\ &\beta(\beta\gamma-(2\delta^{2}+\beta\gamma)\lambda_{0}+c) = 0,\\ &\gamma(2\delta^{2}+\beta\gamma-(2\delta^{2}+\beta\gamma)\lambda_{0}+c) = 0,\\ &\delta(\delta^{2}+\beta\gamma-(2\delta^{2}+\beta\gamma)\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(4.14)

Assume first $\alpha = 0$ . Then, $\alpha+\delta\neq0$ and $\alpha\gamma-\beta\delta = 0$ leads to $\beta = 0$ and $\delta\neq0$ . Therefore, system (4.14) reduces to:

$\begin{equation} \left\{ \begin{aligned} &\gamma(2\delta^{2}-2\delta^{2}\lambda_{0}+c) = 0,\\ &\delta(\delta^{2}-2\delta^{2}\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(4.15)

Then, for Case 1), system (4.14) holds.

Now, let $\alpha\neq0$ . Suppose $\delta = 0$ , from the equations in (4.14) I can derive that $\gamma = 0$ . Then, I have $c = 0$ . Consequently, I have Cases 2) and 3). If $\delta\neq0$ , the equations in (4.14) imply that $\gamma = 0$ . Substituting into the second equation in (4.14) leads to $\beta = 0$ . Then, we have 4). □

4.7. Algebraic Schouten soliton of $G_{7}$

Lemma 21. ^[13] The Ricci tensor $\rho^{B_{1}}$ concerning connection $\nabla^{B_{1}}$ of $(G_{7}, g)$ is given by:

$\begin{equation} \rho^{B_{1}}(e_{i},e_{j}) = \left( \begin{matrix} \alpha^{2} & \beta(\alpha+\delta) & \frac{1}{2}\beta(\delta-\alpha)\\ \beta(\alpha+\delta) & 0 & \delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)\\ \frac{1}{2}\beta(\delta-\alpha) & \delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta) & \beta^{2}-\alpha^{2}-\beta\gamma \end{matrix} \right). \end{equation}$

(4.16)

From this, we derive the following theorem.

Theorem 22. If $(G_{7}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{1}}$ ; then, it fulfills the conditions: $\alpha = 2\delta$ , $\beta = \gamma = 0$ , $c = \frac{\alpha^{2}}{2}-2\alpha^{2}\lambda_{0}$ .

Proof. From (4.16), I have the expression for $Ric^{B_{1}}$ as follows:

$\begin{equation*} Ric^{B_{1}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} \alpha^{2} & \beta(\alpha+\delta) & -\frac{1}{2}\beta(\delta-\alpha)\\ \beta(\alpha+\delta) & 0 & -\delta^{2}-\frac{1}{2}(\beta\gamma+\alpha\delta)\\ \frac{1}{2}\beta(\delta-\alpha) & \delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta) & -\beta^{2}+\alpha^{2}+\beta\gamma \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{1}} = 2\alpha^{2}-\beta^{2}+\beta\gamma$ . Now, I can express $D^{B_{1}}$ as follows:

$\begin{equation*} \left\{ \begin{array}{lll} D^{B_{1}}e_{1} = (\alpha^{2}-(2\alpha^{2}-\beta^{2}+\beta\gamma)\lambda_{0}-c)e_{1}+\beta(\alpha+\delta)e_{2}-\frac{1}{2}\beta(\delta-\alpha)e_{3},\\ D^{B_{1}}e_{2} = \beta(\alpha+\delta)e_{1}-((2\alpha^{2}-\beta^{2}+\beta\gamma)\lambda_{0}+c)e_{2}-(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta))e_{3},\\ D^{B_{1}}e_{3} = \frac{1}{2}\beta(\delta-\alpha)e_{1}+(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta))e_{2}+(\alpha^{2}-\beta^{2}+\beta\gamma-(2\alpha^{2}-\beta^{2}+\beta\gamma)\lambda_{0}-c)e_{3}. \end{array} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{1}}$ on $(G_{7}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{array}{ll} \alpha(s^{B_{1}}\lambda_{0}+c)-\frac{1}{2}\beta(\beta+\gamma)(\delta-\alpha)-\beta^{2}(\alpha+\delta)-\alpha(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)) = 0,\\ \beta(-\alpha^{2}+s^{B_{1}}\lambda_{0}+c)+\frac{1}{2}\beta\delta(\delta-\alpha)+\alpha\beta(\alpha+\delta) = 0,\\ \beta(-\beta^{2}+\beta\gamma+s^{B_{1}}\lambda_{0}+c)+\frac{1}{2}\beta(\delta-\alpha)^{2}-2\beta(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)) = 0,\\ \alpha(\alpha^{2}+\beta\gamma-\beta^{2}-s^{B_{1}}\lambda_{0}-c)+\beta(\gamma-\beta)(\alpha+\delta)-\frac{\beta^{2}(\delta-\alpha)}{2}-\alpha(\delta^{2}+\frac{\beta\gamma+\alpha\delta}{2}) = 0,\\ \beta(2\alpha^{2}+\beta\gamma-\beta^{2}-s^{B_{1}}\lambda_{0}-c)+\beta(\delta-\alpha)(\alpha+\delta)-2\beta(\delta^{2}+\frac{\beta\gamma+\alpha\delta}{2}) = 0,\\ \beta(\alpha^{2}-s^{B_{1}}\lambda_{0}-c)+\beta\delta(\alpha+\delta)+\frac{1}{2}\alpha\beta(\delta-\alpha) = 0,\\ \gamma(-\beta^{2}+\beta\gamma-s^{B_{1}}\lambda_{0}-c)+\beta(\alpha-\delta)(\alpha+\delta)-\frac{1}{2}\beta(\alpha-\delta)^{2} = 0,\\ \delta(\alpha^{2}-\beta^{2}+\beta\gamma-s^{B_{1}}\lambda_{0}-c)+\beta(\beta-\gamma)(\alpha+\delta)+\frac{\beta^{2}(\delta-\alpha)}{2}-\delta(\delta^{2}+\frac{\beta\gamma+\alpha\delta}{2}) = 0,\\ -\delta(s^{B_{1}}\lambda_{0}+c)+\beta^{2}(\alpha+\delta)+\frac{1}{2}\beta(\beta+\gamma)(\delta-\alpha)+\delta(\delta^{2}+\frac{1}{2}(\beta\gamma+\alpha\delta)) = 0. \end{array} \right. \end{equation}$

(4.17)

Since $\alpha\gamma = 0$ and $\alpha+\delta\neq0$ , I now analyze the system under different assumptions.

First, if $\alpha = 0$ and $\gamma\neq0$ , from the equations above, and after simple calculation, we have $\beta = 0$ . Furthermore, the seventh and eighth equations imply that $\delta^{3} = 0$ , which leads to a contradiction.

Second, if $\alpha\neq0$ and $\gamma = 0$ . the seventh equation gives rise to three possible subcases: $\beta = 0$ , $\alpha = \delta$ , or $\alpha+3\delta = 0$ . Initially, let's assume $\beta = 0$ . In this case, the equations in system (4.17), imply that $(\alpha-2\delta)(\alpha+\delta) = 0$ , leading to $\alpha = 2\delta$ , and the theorem is true. Next, I consider the subcase where $\alpha = \delta$ and $\beta\neq0$ . Then, the fifth and sixth equations result in $4\alpha^{2}+\beta^{2} = 0$ , which leads to a contradiction. Additionally, I consider that $\alpha+3\delta = 0$ and $\beta\neq0$ . The first and last equations provide $(2\alpha^{2}-\beta^{2}+\beta\gamma)\lambda_{0}+c = 0$ . Substituting this into the third equation derives $\beta = 0$ , which leads to a contradiction.

Finally, if $\alpha = \gamma = 0$ . The first equation in system (4.17) leads to $\beta = 0$ . Then, using the equations in (4.17), we have $\delta^{3} = 0$ , which leads to a contradiction. □

5. Algebraic Schouten solitons concerning connection $\nabla^{B_{2}}$

In this section, I formulate the algebraic criterion necessary for a three-dimensional Lorentzian Lie group to have an algebraic Schouten soliton related to the given Bott connection $\nabla^{B_{2}}$ . Let us recall the Bott connection with the third distribution, denoted by $\nabla^{B_{2}}$ . Consider a smooth manifold $(M, g)$ that is equipped with the Levi-Civita connection $\nabla$ , and let $TM$ represent its tangent bundle, spanned by $\{e_{1}, e_{2}, e_{3}\}$ . I introduce a distribution $D_{2}$ spanned by $\{e_{2}, e_{3}\}$ and its orthogonal complement $D^{\perp}_{2}$ , which is spanned by $\{e_{1}\}$ . The Bott connection $\nabla^{B}_{2}$ associated with the distribution $D_{2}$ is then defined as follows:

$\begin{equation} \nabla_{X}^{B_{2}}Y = \left\{ \begin{aligned} &\pi_{D_{2}}(\nabla^{LC}_{X}Y),\quad X,Y\in\Gamma^{\infty}(D_{2}),\\ &\pi_{D_{2}}([X,Y]),\quad X\in\Gamma^{\infty}(D_{2}^{\perp}),\; Y\in\Gamma^{\infty}(D_{2}),\\ &\pi_{D_{2}^{\perp}}([X,Y]),\quad X\in\Gamma^{\infty}(D_{2}),\; Y\in\Gamma^{\infty}(D_{2}^{\perp}),\\ &\pi_{D_{2}^{\perp}}(\nabla^{LC}_{X}Y),\quad X,Y\in\Gamma^{\infty}(D_{2}^{\perp}), \end{aligned} \right. \end{equation}$

(5.1)

where $\pi_{D_{2}}$ (respectively, $\pi_{D_{2}^{\perp}}$ ) denotes the projection onto $D_{2}$ (respectively, $D_{2}^{\perp}$ ).

5.1. Algebraic Schouten soliton of $G_{1}$

Lemma 23. ^[13] The Ricci tensor $\rho^{B_{2}}$ concerning connection $\nabla^{B_{2}}$ of $(G_{1}, g)$ is given by:

$\begin{equation} \rho^{B_{2}}(e_{i},e_{j}) = \left( \begin{matrix} 0 & \frac{1}{2}\alpha\beta & -\frac{1}{2}\alpha\beta\\ \frac{1}{2}\alpha\beta & -\beta^{2} & 0\\ -\frac{1}{2}\alpha\beta & 0 & \beta^{2} \end{matrix} \right). \end{equation}$

(5.2)

From this, I derive the following theorem.

Theorem 24. If $(G_{1}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{2}}$ ; then, it fulfills the conditions: $\alpha\neq0$ , $\beta = 0$ and $c = 0$ .

Proof. According to (5.2), the expression for $Ric^{B_{2}}$ is derived as follows:

$\begin{equation*} Ric^{B_{2}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} 0 & \frac{1}{2}\alpha\beta & \frac{1}{2}\alpha\beta\\ \frac{1}{2}\alpha\beta & -\beta^{2} & 0\\ -\frac{1}{2}\alpha\beta & 0 & -\beta^{2} \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{2}} = -2\beta^{2}$ . Now, I can express $D^{B_{2}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{2}}e_{1} = (2\beta^{2}\lambda_{0}-c)e_{1}+\frac{1}{2}\alpha\beta e_{2}+\frac{1}{2}\alpha\beta e_{3},\\ &D^{B_{2}}e_{2} = \frac{1}{2}\alpha\beta e_{1}-(\beta^{2}-2\beta^{2}\lambda_{0}+c)e_{2},\\ &D^{B_{2}}e_{3} = -\frac{1}{2}\alpha\beta e_{1}-(\beta^{2}-2\beta^{2}\lambda_{0}+c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{2}}$ on $(G_{1}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\beta^{2}-2\beta^{2}\lambda_{0}+c)+\alpha\beta^{2} = 0,\\ &\alpha^{2}\beta = 0,\\ &\alpha(-2\beta^{2}\lambda_{0}+c)-\alpha\beta^{2} = 0,\\ &\beta(-2\beta^{2}\lambda_{0}+c)+\alpha^{2}\beta = 0. \end{aligned} \right. \end{equation}$

(5.3)

By solving (5.3), I have $\alpha\neq0$ , $\beta = 0$ and $c = 0$ . □

5.2. Algebraic Schouten soliton of $G_{2}$

Lemma 25. ^[13] The Ricci tensor $\rho^{B_{2}}$ concerning connection $\nabla^{B_{2}}$ of $(G_{2}, g)$ is given by:

$\begin{equation} \rho^{B_{2}}(e_{i},e_{j}) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & -\alpha\beta & -\alpha\gamma\\ 0 & -\alpha\gamma & \alpha\beta \end{matrix} \right). \end{equation}$

(5.4)

From this, I derive the following theorem.

Theorem 26. If $(G_{2}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{2}}$ ; then, one of the following cases holds:

1) $\alpha = 0$ , $\beta = 0$ , $c = 0$ ;

2) $\alpha = 0$ , $\beta\neq0$ , $c = 0$ .

Proof. From (5.4), I have the expression for $Ric^{B_{2}}$ as follows:

$\begin{equation*} Ric^{B_{2}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & -\alpha\beta & \alpha\gamma\\ 0 & -\alpha\gamma & -\alpha\beta \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{2}} = -2\alpha\beta$ . Now, I can express $D^{B_{2}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{2}}e_{1} = (2\alpha\beta\lambda_{0}-c)e_{1},\\ &D^{B_{2}}e_{2} = -(\alpha\beta-2\alpha\beta\lambda_{0}+c)e_{2}+\alpha\gamma e_{3},\\ &D^{B_{2}}e_{3} = -\alpha\gamma e_{2}-(\alpha\beta-2\alpha\beta\lambda_{0}+c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{2}}$ on $(G_{2}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\gamma(-2\alpha\beta\lambda_{0}+c)-2\alpha\beta\gamma = 0,\\ &\beta(-2\alpha\beta\lambda_{0}+c)-2\alpha\gamma^{2} = 0,\\ &\alpha(2\alpha\beta-2\alpha\beta\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(5.5)

Since $\gamma\neq0$ , I assume first that $\alpha = 0$ . Under this assumption, the first two equations in (5.5) yield $c = 0$ . Therefore, Cases 1) and 2) hold. Now, let $\alpha\neq0$ , then I have $\beta = 0$ , and the second equation in (5.5) becomes $2\alpha\gamma^{2} = 0$ , which is a contradiction. □

5.3. Algebraic Schouten soliton of $G_{3}$

Lemma 27. ^[13] The Ricci tensor $\rho^{B_{2}}$ concerning connection $\nabla^{B_{2}}$ of $(G_{3}, g)$ is given by:

$\begin{equation} \rho^{B_{2}}(e_{i},e_{j}) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & \alpha\beta \end{matrix} \right). \end{equation}$

(5.6)

From this, I derive the following theorem.

Theorem 28. If $(G_{3}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{2}}$ ; then, one of the following cases holds:

1) $\alpha = \beta = \gamma = 0$ , for all $c$ ;

2) $\alpha = \gamma = 0$ , $\beta\neq0$ , $c = 0$ ;

3) $\alpha\neq0$ , $\beta = \gamma = 0$ , $c = 0$ ;

4) $\alpha\neq0$ , $\beta\neq0$ , $\gamma = 0$ , $c = -\alpha\beta+\alpha\beta\lambda_{0}$ ;

5) $\alpha = \beta = 0$ , $\gamma\neq0$ , $c = 0$ .

Proof. According to (5.6), the expression for $Ric^{B_{2}}$ is derived as follows:

$\begin{equation*} Ric^{B_{2}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -\alpha\beta \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{2}} = -\alpha\beta$ . Now, I can express $D^{B_{2}}$ as follows:

$\begin{equation} \left\{ \begin{aligned} &D^{B_{2}}e_{1} = (\alpha\beta\lambda_{0}-c)e_{1},\\ &D^{B_{2}}e_{2} = (\alpha\beta\lambda_{0}-c)e_{2},\\ &D^{B_{2}}e_{3} = -(\alpha\beta-\alpha\beta\lambda_{0}+c)e_{3}. \end{aligned} \right. \end{equation}$

(5.7)

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{2}}$ on $(G_{3}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\gamma(-\alpha\beta-\alpha\beta\lambda_{0}+c) = 0,\\ &\beta(\alpha\beta-\alpha\beta\lambda_{0}+c) = 0,\\ &\alpha(\alpha\beta-\alpha\beta\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(5.8)

Assuming that $\gamma = 0$ , I have

$\begin{equation} \left\{ \begin{aligned} &\beta(\alpha\beta-\alpha\beta\lambda_{0}+c) = 0,\\ &\alpha(\alpha\beta-\alpha\beta\lambda_{0}+c) = 0. \end{aligned} \right. \end{equation}$

(5.9)

Then, for Cases 1)–4), system (5.8) holds. Now, let $\gamma\neq0$ , then we have $\alpha = \beta = 0$ . Then, the Case 5) is true. □

5.4. Algebraic Schouten soliton of $G_{4}$

Lemma 29. ^[13] The Ricci tensor $\rho^{B_{2}}$ concerning connection $\nabla^{B_{2}}$ of $(G_{4}, g)$ is given by:

$\begin{equation} \rho^{B_{2}}(e_{i},e_{j}) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & \alpha(2\eta-\beta)& \alpha\\ 0 & \alpha & \alpha\beta \end{matrix} \right). \end{equation}$

(5.10)

From this, I derive the following theorem.

Theorem 30. If $(G_{4}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{2}}$ , then one of the following cases holds:

1) $\alpha = 0$ , $c = 0$ ;

2) $\alpha\neq0$ , $\beta = \eta$ , $c = 0$ .

Proof. From (5.10), I have the expression for $Ric^{B_{2}}$ as follows:

$\begin{equation*} Ric^{B_{2}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & \alpha(2\eta-\beta)& -\alpha\\ 0 & \alpha & -\alpha\beta \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{2}} = \alpha(2\eta-\beta)-\alpha\beta$ . Now, I can express $D^{B_{2}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{2}}e_{1} = -((\alpha(2\eta-\beta)-\alpha\beta)\lambda_{0}+c)e_{1},\\ &D^{B_{2}}e_{2} = (\alpha(2\eta-\beta)-(\alpha(2\eta-\beta)-\alpha\beta)\lambda_{0}-c)e_{2}-\alpha e_{3},\\ &D^{B_{2}}e_{3} = \alpha e_{2}-(\alpha\beta+(\alpha(2\eta-\beta)-\alpha\beta)\lambda_{0}+c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{2}}$ on $(G_{4}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &2\alpha(\eta-\beta)-((\alpha(2\eta-\beta)-\alpha\beta)\lambda_{0}+c) = 0,\\ &(2\eta-\beta)(2\alpha\eta-(\alpha(2\eta-\beta)-\alpha\beta)\lambda_{0}-c)-2\alpha = 0,\\ &\beta(2\alpha\eta+(\alpha(2\eta-\beta)-\alpha\beta)\lambda_{0}+c)-2\alpha = 0,\\ &\alpha(\alpha(2\eta-\beta)-\alpha\beta-(\alpha(2\eta-\beta)-\alpha\beta)\lambda_{0}-c) = 0. \end{aligned} \right. \end{equation}$

(5.11)

Assume first that $\alpha = 0$ , then system (5.11) holds trivially. Therefore, Case 1) holds. Now, let $\alpha\neq0$ I have $\beta = \eta$ ; then, for Case 2), system (5.11) holds. □

5.5. Algebraic Schouten soliton of $G_{5}$

Lemma 31. ^[13] The Ricci tensor $\rho^{B_{2}}$ concerning connection $\nabla^{B_{2}}$ of $(G_{5}, g)$ is given by:

$\begin{equation} \rho^{B_{2}}(e_{i},e_{j}) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & \delta^{2} & 0\\ 0 & 0 & -(\beta\gamma+\delta^{2}) \end{matrix} \right). \end{equation}$

(5.12)

From this, I derive the following theorem.

Theorem 32. If $(G_{5}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{2}}$ , then one of the following cases holds:

i. $\alpha = \beta = \gamma = 0$ , $\delta\neq0$ , $c = \delta^{2}-2\delta^{2}\lambda_{0}$ ;

ii. $\alpha\neq0$ , $\beta = \delta = \gamma = 0$ , $c = 0$ ;

iii. $\alpha\neq0$ , $\beta\neq0$ , $\delta = \gamma = 0$ , $c = 0$ ;

iv. $\alpha\neq0$ , $\beta = \gamma = 0$ , $\delta\neq0$ , $c = \delta^{2}-2\delta^{2}\lambda_{0}$ .

Proof. From (5.12), I have the expression for $Ric^{B_{2}}$ as follows:

$\begin{equation*} Ric^{B_{2}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & \delta^{2} & 0\\ 0 & 0 & \beta\gamma+\delta^{2} \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{2}} = \beta\gamma+2\delta^{2}$ . Now, I can express $D^{B_{2}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{2}}e_{1} = -((\beta\gamma+2\delta^{2})\lambda_{0}+c)e_{1},\\ &D^{B_{2}}e_{2} = (\delta^{2}-(\beta\gamma+2\delta^{2})\lambda_{0}-c)e_{2},\\ &D^{B_{2}}e_{3} = (\beta\gamma+\delta^{2}-(\beta\gamma+2\delta^{2})\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{2}}$ on $(G_{5}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\beta\gamma+\delta^{2}-(\beta\gamma+2\delta^{2})\lambda_{0}-c) = 0,\\ &\beta(\beta\gamma-(\beta\gamma+2\delta^{2})\lambda_{0}-c) = 0,\\ &\gamma(\beta\gamma+2\delta^{2}-(\beta\gamma+2\delta^{2})\lambda_{0}-c) = 0,\\ &\delta(\beta\gamma+\delta^{2}-(\beta\gamma+2\delta^{2})\lambda_{0}-c) = 0. \end{aligned} \right. \end{equation}$

(5.13)

Let $\alpha = 0$ , then I have $\beta = 0$ and $\delta\neq0$ . In this case, (5.13) reduces to:

$\begin{equation} \left\{ \begin{aligned} &\gamma(2\delta^{2}-2\delta^{2}\lambda_{0}-c) = 0,\\ &\delta(\delta^{2}-2\delta^{2}\lambda_{0}-c) = 0. \end{aligned} \right. \end{equation}$

(5.14)

Therefore, I conclude that $\gamma = 0$ , and we have Case i.

Next, I consider the case where $\alpha\neq0$ . By combining the first and third equations from (5.13), we obtain $\gamma\delta^{2} = 0$ . If $\gamma = \delta = 0$ , then $c = 0$ . Therefore, Cases ii and iii hold. If $\gamma = 0$ and $\delta\neq0$ , then $\beta = 0$ . Therefore, Case iv holds. □

5.6. Algebraic Schouten soliton of $G_{6}$

Lemma 33. ^[13] The Ricci tensor $\rho^{B_{2}}$ concerning connection $\nabla^{B_{2}}$ of $(G_{6}, g)$ is given by:

$\begin{equation} \rho^{B_{2}}(e_{i},e_{j}) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{matrix} \right). \end{equation}$

(5.15)

From this, I derive the following theorem.

Theorem 34. If $(G_{6}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{2}}$ , then we have $c = 0$ .

Proof. According to (5.15), the expression for $Ric^{B_{2}}$ is derived as follows:

$\begin{equation*} Ric^{B_{2}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{2}} = 0$ . Now, I can express $D^{B_{2}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{2}}e_{1} = -ce_{1},\\ &D^{B_{2}}e_{2} = -ce_{2},\\ &D^{B_{2}}e_{3} = -ce_{3}. \end{aligned} \right. \end{equation*}$

Hence, by (2.4), I have $c = 0$ . □

5.7. Algebraic Schouten soliton of $G_{7}$

Lemma 35. ^[13] The Ricci tensor $\rho^{B_{2}}$ concerning connection $\nabla^{B_{2}}$ of $(G_{7}, g)$ is given by:

$\begin{equation} \rho^{B_{2}}(e_{i},e_{j}) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & -\beta\gamma & \beta\gamma\\ 0 & \beta\gamma & -\beta\gamma \end{matrix} \right). \end{equation}$

(5.16)

From this, I derive the following theorem.

Theorem 36. If $(G_{7}, g)$ constitutes an algebraic Schouten soliton concerning connection $\nabla^{B_{2}}$ , then one of the following cases holds:

1) $\alpha = \beta = 0$ , $\gamma\neq0$ , $c = 0$ ;

2) $\alpha\neq0$ , $\gamma = 0$ , $c = 0$ ;

3) $\alpha = \gamma = 0$ , $c = 0$ .

Proof. From (5.16), I have the expression for $Ric^{B_{2}}$ as follows:

$\begin{equation*} Ric^{B_{2}}\left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right) = \left( \begin{matrix} 0 & 0 & 0\\ 0 & -\beta\gamma & -\beta\gamma\\ 0 & \beta\gamma & \beta\gamma \end{matrix} \right) \left( \begin{matrix} e_{1}\\ e_{2}\\ e_{3} \end{matrix}\right). \end{equation*}$

The scalar curvature is $s^{B_{2}} = 0$ . Now, I can express $D^{B_{2}}$ as follows:

$\begin{equation*} \left\{ \begin{aligned} &D^{B_{2}}e_{1} = -(s\lambda_{0}+c)e_{1},\\ &D^{B_{2}}e_{2} = -(\beta\gamma+s\lambda_{0}+c)e_{2}-\beta\gamma e_{3},\\ &D^{B_{2}}e_{3} = \beta\gamma e_{2}+(\beta\gamma-s\lambda_{0}-c)e_{3}. \end{aligned} \right. \end{equation*}$

Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with $\nabla^{B_{2}}$ on $(G_{7}, g)$ , if and only if the following condition satisfies:

$\begin{equation} \left\{ \begin{aligned} &\alpha(\beta\gamma+c)-\alpha\beta\gamma = 0,\\ &\beta c = 0,\\ &\beta(2\beta\gamma+c)-2\beta^{2}\gamma = 0,\\ &\alpha(\beta\gamma-c) = \alpha\beta\gamma,\\ &\beta c+2\beta^{2}\gamma = 0,\\ &\gamma c = 0,\\ &\delta(-\beta\gamma+c)+\beta\gamma\delta = 0,\\ &\delta(\beta\gamma+c)-\beta\gamma\delta = 0. \end{aligned} \right. \end{equation}$

(5.17)

Since $\alpha\gamma = 0$ and $\alpha+\delta\neq0$ , I now analyze the system under different assumptions.

First, if $\alpha = 0$ and $\gamma\neq0$ , under this assumption, the fifth and sixth equations of (5.17) jointly imply that $\beta = c = 0$ . Therefore, Case 1) holds.

Second, if $\alpha\neq0$ and $\gamma = 0$ , then the first equation of (5.17) gives $c = 0$ , and for Case 2), system (5.17) holds.

Finally, if $\alpha = \gamma = 0$ , then $\delta\neq0$ , and the last equation of (5.17) gives $c = 0$ . Therefore, Case 3) holds. □

6. Conclusions

I present algebraic conditions for three-dimensional Lorentzian Lie groups to be an algebraic Schouten soliton associated with the Bott connection, considering three distributions. The main result indicates that $G_{4}$ and $G_{7}$ do not have such solitons with the first distribution, while the result for $G_{5}$ with the first distribution is trivial, and the other cases all possess algebraic Schouten solitons. In the future, we will explore algebraic Schouten solitons in higher dimensions, as in ^[24,25].

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors deeply appreciate the anonymous reviewers for their insightful feedback and valuable suggestions, greatly enhancing our paper.

Conflict of interest

The authors declare there is no conflicts of interest.

References

[1]	S. Li, H. Li, Q. Zhou, F. Zhang, N. Desneux, S. Wang, et al., Essential oils from two aromatic plants repel the tobacco whitefly Bemisia tabaci, J. Pest Sci., 95 (2022), 971–982. https://doi.org/10.1007/s10340-021-01412-0 doi: 10.1007/s10340-021-01412-0
[2]	A. Cuthbertson, L. F. Blackburn, P. Northing, W. Luo, R. Cannon, K. Walters, Leaf dipping as an environmental screening measure to test chemical efficacy against Bemisia tabaci on poinsettia plants, Int. J. Environ. Sci. Technol., 6 (2009), 347–352. https://doi.org/10.1007/bf03326072 doi: 10.1007/bf03326072
[3]	M. Ahmad, M. I. Arif, Z. Ahmad, I. Denholm, Cotton whitefly (Bemisia tabaci) resistance to organophosphate and pyrethroid insecticides in Pakistan, Pest Manage. Sci., 58 (2002), 203–208. https://doi.org/10.1002/ps.440 doi: 10.1002/ps.440
[4]	B. Xia, Z. Zou, P. Li, P. Lin, Effect of temperature on development and reproduction of Neoseiulus barkeri (Acari: Phytoseiidae) fed on Aleuroglyphus ovatus, Exp. Appl. Acarol., 56 (2012), 33–41. https://doi.org/10.1007/s10493-011-9481-1 doi: 10.1007/s10493-011-9481-1
[5]	Y. Y. Li, M. X. Liu, H. W. Zhou, C. Tian, G. Zhang, Y. Liu, et al., Evaluation of Neoseiulus barkeri (Acari: Phytoseiidae) for control of Eotetranychus kankitus (Acari: Tetranychidae), J. Econ. Entomol., 110 (2017), 903–914. https://doi.org/10.1093/jee/tox056 doi: 10.1093/jee/tox056
[6]	Y. Y. Li, G. H. Zhang, C. B. Tian, M. X. Liu, Y. Liu, H. Liu, et al., Does long-term feeding on alternative prey affect the biological performance of Neoseiulus barkeri (Acari: Phytoseiidae) on the target spider mites, J. Econ. Entomol., 110 (2017), 915–923. https://doi.org/10.1093/jee/tox055 doi: 10.1093/jee/tox055
[7]	Y. Y. Li, J. G. Yuan, M. X. Liu, Z. Zhang, H. Zhou, H. Liu, Evaluation of four artificial diets on demography parameters of Neoseiulus barkeri, BioControl, 66 (2020), 789–802. https://doi.org/10.1007/s10526-021-10108-4 doi: 10.1007/s10526-021-10108-4
[8]	Y. Fan, F. L. Petitt, Functional response of Neoseiulus barkeri Hughes on two-spotted spider mite (Acari: Tetranychidae), Exp. Appl. Acarol., 18 (1994), 613–621. https://doi.org/10.1007/bf00051724 doi: 10.1007/bf00051724
[9]	T. Zou, Effect of photoperiod on development and demographic parameters of Neoseiulus barkeri (Acari: Phytoseiidae) fed on Tyrophagus putrescentiae (Acari: Acaridae), Exp. Appl. Acarol., 70 (2016), 45–56. https://doi.org/10.1007/s10493-016-0065-y doi: 10.1007/s10493-016-0065-y
[10]	H. Yao, W. Zheng, K. Tariq, H. Zhang, Functional and numerical responses of three species of predatory phytoseiid mites (Acari: Phytoseiidae) to Thrips flavidulus (Thysanoptera: Thripidae), Neotrop. Entomol., 43 (2014), 437–445. https://doi.org/10.1007/s13744-014-0229-6 doi: 10.1007/s13744-014-0229-6
[11]	C. Xiang, J. C. Huang, S. G. Ruan, D. M. Xiao, Bifurcation analysis in a host-generalist parasitoid model with Holling Ⅱ functional response, J. Differ. Equations, 268 (2020), 4618–4662. https://doi.org/10.1016/j.jde.2019.10.036 doi: 10.1016/j.jde.2019.10.036
[12]	É. Diz-Pita, M. V. Otero-Espinar, Predator-prey models: A review of some recent advances, Mathematics, 9 (2021), 1783. https://doi.org/10.3390/math9151783 doi: 10.3390/math9151783
[13]	M. X. Chen, R. C. Wu, X. H. Wang, Non-constant steady states and Hopf bifurcation of a species interaction model, Commun. Nonlinear Sci. Numer. Simul., 116 (2023), 106846. https://doi.org/10.1016/j.cnsns.2022.106846 doi: 10.1016/j.cnsns.2022.106846
[14]	M. X. Chen, H. M. Srivastava, Existence and stability of bifurcating solution of a chemotaxis model, Proc. Am. Math. Soc., 151 (2023), 4735–4749. https://doi.org/10.1090/proc/16536 doi: 10.1090/proc/16536
[15]	M. X. Chen, R. C. Wu. Steady states and spatiotemporal evolution of a diffusive predator-prey model, Chaos Solitons Fractals, 170 (2023), 113397. https://doi.org/10.1016/j.chaos.2023.113397 doi: 10.1016/j.chaos.2023.113397
[16]	A. J. Lotka, Eelements of physical biology, Am. J. Public Health, 21 (1926), 341–343. https://doi.org/10.2307/2298330 doi: 10.2307/2298330
[17]	V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558–560. https://doi.org/10.1038/119012b0 doi: 10.1038/119012b0
[18]	D. Ludwig, D. D. Johns, C. S. Holling, Qualitative analysis of insect outbreak system: the spruce budworm and forest, J. Anim. Ecol., 47 (1978), 315–332. https://doi.org/10.2307/3939 doi: 10.2307/3939
[19]	C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5–60. https://doi.org/10.4039/entm9745fv doi: 10.4039/entm9745fv
[20]	Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389–406. https://doi.org/10.1007/s002850050105 doi: 10.1007/s002850050105
[21]	J. Zhou, C. L. Mu, Coexistence states of a Holling type-Ⅱ predator-prey system, J. Math. Anal. Appl., 369 (2010), 555–563. https://doi.org/10.1016/j.jmaa.2010.04.001 doi: 10.1016/j.jmaa.2010.04.001
[22]	K. Vishwakarma, M. Sen, Role of Allee effect in prey and hunting cooperation in a generalist predator, Math. Comput. Simul., 190 (2021), 622–640. https://doi.org/10.1016/j.matcom.2021.05.023 doi: 10.1016/j.matcom.2021.05.023
[23]	M. Lu, J. C. Huang, Global analysis in Bazykin's model with Holling Ⅱ functional response and predator competition, J. Differ. Equations, 280 (2021), 99–138. https://doi.org/10.1016/j.jde.2021.01.025 doi: 10.1016/j.jde.2021.01.025
[24]	Y. Tian, H. M. Li, The study of a predator-prey model with fear effect based on state-dependent harvesting strategy, Complexity, 2022 (2022). https://doi.org/10.1155/2022/9496599 doi: 10.1155/2022/9496599
[25]	M. X. Chen, X. Z. Li, R. C. Wu, Bifurcations and steady states of a predator-prey model with strong Allee and fear effects, Int. J. Biomath., 2023 (2023). https://doi.org/10.1142/s1793524523500663 doi: 10.1142/s1793524523500663
[26]	H. Li, Y. Tian, Dynamic behavior analysis of a feedback control predator-prey model with exponential fear effect and Hassell-Varley functional response, J. Franklin Inst., 360 (2023), 3479–3498. https://doi.org/10.1016/j.jfranklin.2022.11.030 doi: 10.1016/j.jfranklin.2022.11.030
[27]	J. H. P. Dawes, M. O. Souza, A derivation of Holling's type Ⅰ, Ⅱ and Ⅲ functional responses in predator-prey systems, J. Theor. Biol., 327 (2013), 11–22. https://doi.org/10.1016/j.jtbi.2013.02.017 doi: 10.1016/j.jtbi.2013.02.017
[28]	Z. J. Liu, S. M. Zhong, C. Yin, W. F. Chen, Dynamics of impulsive reaction-diffusion predator-prey system with Holling type Ⅲ functional response, Appl. Math. Modell., 35 (2011), 5564–5578. https://doi.org/10.1016/j.apm.2011.05.019 doi: 10.1016/j.apm.2011.05.019
[29]	J. C. Huang, S. G. Ruan, R. J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Differ. Equations, 257 (2014), 1721–1752. https://doi.org/10.1016/j.jde.2014.04.024 doi: 10.1016/j.jde.2014.04.024
[30]	Y. F. Dai, Y. L. Zhao, B. Sang, Four limit cycles in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, Nonlinear Anal. Real World Appl., 50 (2019), 218–239. https://doi.org/10.1016/j.nonrwa.2019.04.003 doi: 10.1016/j.nonrwa.2019.04.003
[31]	Q. Yang, X. H. Zhang, D. Q. Jiang, M. G. Shao, Analysis of a stochastic predator-prey model with weak Allee effect and Holling-(n+1) functional response. Commun. Nonlinear Sci. Numer. Simul., 111 (2022), 106454. https://doi.org/10.1016/j.cnsns.2022.106454 doi: 10.1016/j.cnsns.2022.106454
[32]	P. D. N. Srinivasu, B. Prasad, M. Venkatesulu, Biological control through provision of additional food to predators: A theoretical study, Theor. Popul Biol., 72 (2007), 111–120. https://doi.org/10.1016/j.tpb.2007.03.011 doi: 10.1016/j.tpb.2007.03.011
[33]	M. R. Wade, M. P. Zalucki, S. D. Wrateen, K. A. Robinson, Conservation biological control of arthropods using artificial food sprays: Current staus and future challenges, Biol. Control, 45 (2008), 185–199. https://doi.org/10.1016/j.biocontrol.2007.10.024 doi: 10.1016/j.biocontrol.2007.10.024
[34]	P. N. G. Srinivasu, B. Prasad, Time optimal control of an additional food provided predator-prey system with applications to pest management and biological conservation, J. Math. Biol., 60 (2010), 591–613. https://doi.org/10.1007/s00285-009-0279-2 doi: 10.1007/s00285-009-0279-2
[35]	P. D. N. Srinivasu, B. Prasad, Role of quantity of additional food to predators as a control in predator-prey systems with relevance to pest management and biological conservation, Bull. Math. Biol., 73 (2011), 2249–2276. https://doi.org/10.1007/s11538-010-9601-9 doi: 10.1007/s11538-010-9601-9
[36]	P. D. N. Srinivasu, D. K. K. Vamsi, V. S. Ananth, Additional food supplements as a tool for biological conservation of predator-prey systems involving type Ⅲ functional response: A qualitative and quantitative investigation, J. Theor. Biol., 455 (2018), 303–318. https://doi.org/10.1016/j.jtbi.2018.07.019 doi: 10.1016/j.jtbi.2018.07.019
[37]	H. J. Barclay, Models for pest control using predator release, habitat management and pesticide release in combination, J. Appl. Ecol., 19 (1982), 337–348. https://doi.org/10.2307/2403471 doi: 10.2307/2403471
[38]	J. C. Van Lenteren, J. Woets, Biological and integrated pest control in greenhouses, Annu. Rev. Entomol., 33 (1988), 239–269. https://doi.org/10.1146/annurev.en.33.010188.001323 doi: 10.1146/annurev.en.33.010188.001323
[39]	P. S. Simenov, D. D. Bainov, Orbital stability of the periodic solutions of autonomous systems with impulse effect, Int. J. Syst. Sci., 19 (1988), 2561–2585. https://doi.org/10.1080/00207728808547133 doi: 10.1080/00207728808547133
[40]	S. Y. Tang, L. S. Chen, Modelling and analysis of integrated pest management strategy, Discrete Contin. Dyn. Syst. B, 4 (2004), 759–768. https://doi.org/10.3934/dcdsb.2004.4.759 doi: 10.3934/dcdsb.2004.4.759
[41]	B. Liu, Y. Zhang, L. Chen, The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Anal. Real World Appl., 6 (2005), 227–243. https://doi.org/10.1016/j.nonrwa.2004.08.001 doi: 10.1016/j.nonrwa.2004.08.001
[42]	H. Zhang, J. J. Jiao, L. S. Chen, Pest management through continuous and impulsive control strategies, Biosystems, 90 (2007), 350–361. https://doi.org/10.1016/j.biosystems.2006.09.038 doi: 10.1016/j.biosystems.2006.09.038
[43]	Y. Z. Pei, X. H. Ji, C. G. Li, Pest regulation by means of continuous and impulsive nonlinear controls, Math. Comput. Modell., 51 (2010), 810–822. https://doi.org/10.1016/j.mcm.2009.10.013 doi: 10.1016/j.mcm.2009.10.013
[44]	X. Y. Song, Y. F. Li, Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type Ⅱ schemes and impulsive effect, Nonlinear Anal. Real World Appl., 9 (2008), 64–79. https://doi.org/10.1016/j.nonrwa.2006.09.004 doi: 10.1016/j.nonrwa.2006.09.004
[45]	X. Q. Wang, W. M. Wang, X. L. Lin, Dynamics of a periodic Watt-type predator-prey system with impulsive effect, Chaos Solitons Fractals, 39 (2009), 1270–1282. https://doi.org/10.1016/j.chaos.2007.06.031 doi: 10.1016/j.chaos.2007.06.031
[46]	L. N. Qian, Q. S. Lu, Q. G. Meng, Z. S. Feng, Dynamical behaviors of a prey-predator system with impulsive control, J. Math. Anal. Appl., 363 (2010), 345–356. https://doi.org/10.1016/j.jmaa.2009.08.048 doi: 10.1016/j.jmaa.2009.08.048
[47]	S. Y. Tang, G. Y. Tang, R. A. Cheke, Optimum timing for integrated pest management: modelling rates of pesticide application and natural enemy releases, J. Theor. Biol., 264 (2010), 623–638. https://doi.org/10.1016/j.jtbi.2010.02.034 doi: 10.1016/j.jtbi.2010.02.034
[48]	C. Li, S. Tang, The effects of timing of pulse spraying and releasing periods on dynamics of generalized predator-prey model, Int. J. Biomath., 5 (2012), 1250012. https://doi.org/10.1142/s1793524511001532 doi: 10.1142/s1793524511001532
[49]	Y. Z. Pei, M. M. Chen, X. Y. Liang, C. Li, M. Zhu, Optimizing pulse timings and amounts of biological interventions for a pest regulation model, Nonlinear Anal. Hybrid Syst., 27 (2018), 353–365. https://doi.org/10.1016/j.nahs.2017.10.003 doi: 10.1016/j.nahs.2017.10.003
[50]	J. Hui, D. M. Zhu, Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos Solitons Fractals, 29 (2006), 233–251. https://doi.org/10.1016/j.chaos.2005.08.025 doi: 10.1016/j.chaos.2005.08.025
[51]	W. Gao, S. Y. Tang, The effects of impulsive releasing methods of natural enemies on pest control and dynamical complexity, Nonlinear Anal. Hybrid Syst., 5 (2011), 540–553. https://doi.org/10.1016/j.nahs.2010.12.001 doi: 10.1016/j.nahs.2010.12.001
[52]	Z. Y. Xiang, S. Y. Tang, C. C. Xiang, J. H. Wu, On impulsive pest control using integrated intervention strategies, Appl. Math. Comput., 269 (2015), 930–946. https://doi.org/10.1016/j.amc.2015.07.076 doi: 10.1016/j.amc.2015.07.076
[53]	S. Tang, C. Li, B. Tang, X. Wang, Global dynamics of a nonlinear state-dependent feedback control ecological model with a multiple-hump discrete map, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 104900. https://doi.org/10.1016/j.cnsns.2019.104900 doi: 10.1016/j.cnsns.2019.104900
[54]	Q. Zhang, B. Tang, T. Cheng, S. Tang, Bifurcation analysis of a generalized impulsive Kolmogorov model with applications to pest and disease control, SIAM J. Appl. Math. 80 (2020), 1796–1819. https://doi.org/10.1137/19m1279320 doi: 10.1137/19m1279320
[55]	Q. Zhang, S. Tang, Bifurcation analysis of an ecological model with nonlinear state-dependent feedback control by Poincaré map defined in phase set, Commun. Nonlinear Sci. Numer. Simul. 108 (2022), 106212. https://doi.org/10.1016/j.cnsns.2021.106212 doi: 10.1016/j.cnsns.2021.106212
[56]	Q. Zhang, S. Tang, X. Zou, Rich dynamics of a predator-prey system with state-dependent impulsive controls switching between two means, J. Differ. Equations, 364 (2023), 336–377. https://doi.org/10.1016/j.jde.2023.03.030 doi: 10.1016/j.jde.2023.03.030
[57]	Y. Tian, Y. Gao, K. B. Sun, Global dynamics analysis of instantaneous harvest fishery model guided by weighted escapement strategy, Chaos Solitons Fractals, 164 (2022), 112597. https://doi.org/10.1016/j.chaos.2022.112597 doi: 10.1016/j.chaos.2022.112597
[58]	Y. Tian, Y. Gao, K. B. Sun, A fishery predator-prey model with anti-predator behavior and complex dynamics induced by weighted fishing strategies, Math. Biosci. Eng., 20 (2003), 1558–1579. https://doi.org/10.3934/mbe.2023071 doi: 10.3934/mbe.2023071
[59]	Y. Tian, Y. Gao, K. B. Sun, Qualitative analysis of exponential power rate fishery model and complex dynamics guided by a discontinuous weighted fishing strategy, Commun. Nonlinear Sci. Numer. Simul., 118 (2023), 107011. https://doi.org/10.1016/j.cnsns.2022.107011 doi: 10.1016/j.cnsns.2022.107011
[60]	Y. Tian, H. Guo, K. B. Sun, Complex dynamics of two prey-predator harvesting models with prey refuge and interval-valued imprecise parameters, Math. Meth. Appl. Sci., 46 (2023), 14278–14298. https://doi.org/10.1002/mma.9319 doi: 10.1002/mma.9319

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Effects of additional food availability and pulse control on the dynamics of a Holling-(p p +1) type pest-natural enemy model

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Abstract

1. Introduction

2. Preliminaries

3. Algebraic Schouten soliton concerning connection ∇B \nabla^{B}

3.1. Algebraic Schouten soliton of G1 G_{1}

3.2. Algebraic Schouten soliton of G2 G_{2}

3.3. Algebraic Schouten soliton of G3 G_{3}

3.4. Algebraic Schouten soliton of G4 G_{4}

3.5. Algebraic Schouten soliton of G5 G_{5}

3.6. Algebraic Schouten soliton of G6 G_{6}

3.7. Algebraic Schouten soliton of G7 G_{7}

4. Algebraic Schouten solitons concerning connection ∇B1 \nabla^{B_{1}}

4.1. Algebraic Schouten soliton of G1 G_{1}

4.2. Algebraic Schouten soliton of G2 G_{2}

4.3. Algebraic Schouten soliton of G3 G_{3}

4.4. Algebraic Schouten soliton of G4 G_{4}

4.5. Algebraic Schouten soliton of G5 G_{5}

4.6. Algebraic Schouten soliton of G6 G_{6}

4.7. Algebraic Schouten soliton of G7 G_{7}

5. Algebraic Schouten solitons concerning connection ∇B2 \nabla^{B_{2}}

5.1. Algebraic Schouten soliton of G1 G_{1}

5.2. Algebraic Schouten soliton of G2 G_{2}

5.3. Algebraic Schouten soliton of G3 G_{3}

5.4. Algebraic Schouten soliton of G4 G_{4}

5.5. Algebraic Schouten soliton of G5 G_{5}

5.6. Algebraic Schouten soliton of G6 G_{6}

5.7. Algebraic Schouten soliton of G7 G_{7}

6. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

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Effects of additional food availability and pulse control on the dynamics of a Holling-( $p$ +1) type pest-natural enemy model

3. Algebraic Schouten soliton concerning connection $\nabla^{B}$

3.1. Algebraic Schouten soliton of $G_{1}$

3.2. Algebraic Schouten soliton of $G_{2}$

3.3. Algebraic Schouten soliton of $G_{3}$

3.4. Algebraic Schouten soliton of $G_{4}$

3.5. Algebraic Schouten soliton of $G_{5}$

3.6. Algebraic Schouten soliton of $G_{6}$

3.7. Algebraic Schouten soliton of $G_{7}$

4. Algebraic Schouten solitons concerning connection $\nabla^{B_{1}}$

4.1. Algebraic Schouten soliton of $G_{1}$

4.2. Algebraic Schouten soliton of $G_{2}$

4.3. Algebraic Schouten soliton of $G_{3}$

4.4. Algebraic Schouten soliton of $G_{4}$

4.5. Algebraic Schouten soliton of $G_{5}$

4.6. Algebraic Schouten soliton of $G_{6}$

4.7. Algebraic Schouten soliton of $G_{7}$

5. Algebraic Schouten solitons concerning connection $\nabla^{B_{2}}$

5.1. Algebraic Schouten soliton of $G_{1}$

5.2. Algebraic Schouten soliton of $G_{2}$

5.3. Algebraic Schouten soliton of $G_{3}$

5.4. Algebraic Schouten soliton of $G_{4}$

5.5. Algebraic Schouten soliton of $G_{5}$

5.6. Algebraic Schouten soliton of $G_{6}$

5.7. Algebraic Schouten soliton of $G_{7}$